1- Tin hieu va he thong.pdf

Chapter 2Signal and Linear System Analysis

Contents

2.1 Signal Models . . . . . . . . . . . . . . . . . . . . . . 2-32.1.1 Deterministic and Random Signals . . . . . . . . 2-3

2.1.2 Periodic and Aperiodic Signals . . . . . . . . . . 2-3

2.1.3 Phasor Signals and Spectra . . . . . . . . . . . . 2-4

2.1.4 Singularity Functions . . . . . . . . . . . . . . . 2-7

2.2 Signal Classifications . . . . . . . . . . . . . . . . . . 2-112.3 Generalized Fourier Series . . . . . . . . . . . . . . . 2-142.4 Fourier Series . . . . . . . . . . . . . . . . . . . . . . 2-20

2.4.1 Complex Exponential Fourier Series . . . . . . . 2-20

2.4.2 Symmetry Properties of the Fourier Coefficients 2-23

2.4.3 Trigonometric Form . . . . . . . . . . . . . . . 2-25

2.4.4 Parseval’s Theorem . . . . . . . . . . . . . . . . 2-26

2.4.5 Line Spectra . . . . . . . . . . . . . . . . . . . 2-26

2.4.6 Numerical Calculation of Xn . . . . . . . . . . . 2-31

2.4.7 Other Fourier Series Properties . . . . . . . . . . 2-37

2.5 Fourier Transform . . . . . . . . . . . . . . . . . . . . 2-382.5.1 Amplitude and Phase Spectra . . . . . . . . . . 2-39

2-1

CHAPTER 2. SIGNAL AND LINEAR SYSTEM ANALYSIS

2.5.2 Symmetry Properties . . . . . . . . . . . . . . . 2-39

2.5.3 Energy Spectral Density . . . . . . . . . . . . . 2-40

2.5.4 Transform Theorems . . . . . . . . . . . . . . . 2-42

2.5.5 Fourier Transforms in the Limit . . . . . . . . . 2-51

2.5.6 Fourier Transforms of Periodic Signals . . . . . 2-53

2.5.7 Poisson Sum Formula . . . . . . . . . . . . . . 2-59

2.6 Power Spectral Density and Correlation . . . . . . . . 2-602.6.1 The Time Average Autocorrelation Function . . 2-61

2.6.2 Power Signal Case . . . . . . . . . . . . . . . . 2-62

2.6.3 Properties of R.�/ . . . . . . . . . . . . . . . . 2-63

2.7 Linear Time Invariant (LTI) Systems . . . . . . . . . 2-702.7.1 Stability . . . . . . . . . . . . . . . . . . . . . . 2-72

2.7.2 Transfer Function . . . . . . . . . . . . . . . . . 2-72

2.7.3 Causality . . . . . . . . . . . . . . . . . . . . . 2-73

2.7.4 Properties of H.f / . . . . . . . . . . . . . . . . 2-74

2.7.5 Response to Periodic Inputs . . . . . . . . . . . 2-78

2.7.6 Distortionless Transmission . . . . . . . . . . . 2-78

2.7.7 Group and Phase Delay . . . . . . . . . . . . . . 2-79

2.7.8 Nonlinear Distortion . . . . . . . . . . . . . . . 2-83

2.7.9 Ideal Filters . . . . . . . . . . . . . . . . . . . . 2-85

2.7.10 Realizable Filters . . . . . . . . . . . . . . . . . 2-87

2.7.11 Pulse Resolution, Risetime, and Bandwidth . . . 2-91

2.8 Sampling Theory . . . . . . . . . . . . . . . . . . . . . 2-972.9 The Hilbert Transform . . . . . . . . . . . . . . . . . 2-972.10 The Discrete Fourier Transform and FFT . . . . . . . 2-97

2-2 ECE 5625 Communication Systems I

2.1. SIGNAL MODELS

2.1 Signal Models

2.1.1 Deterministic and Random Signals

� Deterministic Signals, used for this course, can be modeled ascompletely specified functions of time, e.g.,

x.t/ D A.t/ cosŒ2�f0.t/t C �.t/�

– Note that here we have also made the amplitude, fre-quency, and phase functions of time

– To be deterministic each of these functions must be com-pletely specified functions of time

� Random Signals, used extensively in Comm Systems II, takeon random values with known probability characteristics, e.g.,

x.t/ D x.t; �i/

where �i corresponds to an elementary outcome from a samplespace in probability theory

– The �i create a ensemble of sample functions x.t; �i/, de-pending upon the outcome drawn from the sample space

2.1.2 Periodic and Aperiodic Signals

� A deterministic signal is periodic if we can write

x.t C nT0/ D x.t/

for any integer n, with T0 being the signal fundamental period

ECE 5625 Communication Systems I 2-3


� A signal is aperiodic otherwise, e.g.,

….t/ D

(1; jt j � 1=2

0; otherwise

(a) periodic signal, (b) aperiodic signal, (c) random signal

2.1.3 Phasor Signals and Spectra

� A complex sinusoid can be viewed as a rotating phasor

Qx.t/ D Aej.!0tC�/; �1 < t <1

� This signal has three parameters, amplitude A, frequency !0,and phase �

� The fixed phasor portion is Aej� while the rotating portion isej!0t


2.1. SIGNAL MODELS

� This signal is periodic with period T0 D 2�=!0

� It also related to the real sinusoid signal A cos.!0t C �/ viaEuler’s theorem

x.t/ D Re˚Qx.t/

D Re

˚A cos.!0t C �/C jA sin.!0t C �/

D A cos.!0t C �/

(a) obtain x.t/ from Qx.t/, (b) obtain x.t/ from Qx.t/ and Qx�.t/

� We can also turn this around using the inverse Euler formula

x.t/ D A cos.!0t C �/

D1

2Qx.t/C

1

2Qx�.t/

DAej.!0tC�/ C Ae�j.!0tC�/

2

� The frequency spectra of a real sinusoid is the line spectra plot-ted in terms of the amplitude and phase versus frequency



� The relevant parameters are A and � for a particular f0 D!0=.2�/

(a) Single-sided line spectra, (b) Double-sided line spectra

� Both the single-sided and double-sided line spectra, shownabove, correspond to the real signal x.t/ D A cos.2�f0t C �/

Example 2.1: Multiple Sinusoids

� Suppose that

x.t/ D 4 cos.2�.10/t C �=3/C 24 sin.2�.100/t � �=8/

� Find the two-sided amplitude and phase line spectra of x.t/

� First recall that cos.!0t � �=2/ D sin.!0t /, so

x.t/ D 4 cos.2�.10/t C �=3/C 24 cos.2�.100/t � 5�=8/

� The complex sinusoid form is directly related to the two-sidedline spectra since each real sinusoid is composed of positiveand negative frequency complex sinusoids

x.t/ D 2hej.2�.10/tC�=3/ C e�j.2�.10/tC�=3/

iC 12

hej.2�.100/t�5�=8/ C e�j.2�.100/t�5�=8/

i2-6 ECE 5625 Communication Systems I

2.1. SIGNAL MODELS

f (Hz)

f (Hz)

Am

plitu

dePh

ase

10

12

2

100-100 -10

5π/8

-5π/8-π/3

π/3

Two-sided amplitude and phase line spectra

2.1.4 Singularity Functions

Unit Impulse (Delta) Function

� Singularity functions, such as the delta function and unit step

� The unit impulse function, ı.t/ has the operational propertiesZ t2

t1

ı.t � t0/ dt D 1; t1 < t0 < t2

ı.t � t0/ D 0; t ¤ t0

which implies that for x.t/ continuous at t D t0, the siftingproperty holds Z

1

�1

x.t/ı.t � t0/ dt D x.t0/

– Alternatively the unit impulse can be defined asZ1

�1

x.t/ı.t/ dt D x.0/



� Properties:

1. ı.at/ D ı.t/=jaj

2. ı.�t / D ı.t/

3. Sifting property special cases

Z t2

t1

x.t/ı.t � t0/ dt D

8̂̂<̂:̂x.t0/; t1 < t0 < t2

0; otherwise

undefined; t0 D t1 or t0 D t2

4. Sampling property

x.t/ı.t � t0/ D x.t0/ı.t � t0/

for x.t/ continuous at t D t05. Derivative propertyZ t2

t1

x.t/ı.n/.t � t0/ dt D .�1/nx.n/.t0/

D .�1/nd nx.t/

dtn

ˇ̌̌̌tDt0

Note: Dealing with the derivative of a delta function re-quires care

� A test function for the unit impulse function helps our intuitionand also helps in problem solving

� Two functions of interest are

ı�.t/ D1

2�…

�t

2�

�D

(12�; jt j � �

0; otherwise

ı1�.t/ D �

�1

�tsin�t

�

�22-8 ECE 5625 Communication Systems I

2.1. SIGNAL MODELS

Test functions for the unit impulse ı.t/: (a) ı�.t/, (b) ı1�.t/

� In both of the above test functions letting � ! 0 results in afunction having the properties of a true delta function

Unit Step Function

� The unit step function can be defined in terms of the unit im-pulse

u.t/ �

Z t

�1

ı.�/ d� D

8̂̂<̂:̂0; t < 0

1; t > 0

undefined; t D 0

alsoı.t/ D

du.t/

dt



Example 2.2: Unit Impulse 1st-Derivative

� Consider Z1

�1

x.t/ı0.t/ dt

� Using the rectangular pulse test function, ı�.t/, we note that

ı�.t/ D1

2�…

�t

2�

�alsoD

1

2�

�u.t C �/ � u.t � �/

�and

dı�.t/

dtD

1

2�

�ı.t C �/ � ı.t � �/

�� Placing the above in the integrand with x.t/ we obtain, with

the aid of the sifting property, thatZ1

�1

x.t/ı0.t/ dt D lim�!0

1

2�

�x.t C �/ � x.t � �/

�D lim

�!0

��x.t � �/ � x.t C �/

�2�

D �x0.0/


2.2. SIGNAL CLASSIFICATIONS

2.2 Signal Classifications

� From circuits and systems we know that a real voltage or cur-rent waveform, e.t/ or i.t/ respectively, measured with respec-tive a real resistance R, the instantaneous power is

P.t/ D e.t/i.t/ D i2.t/R W

� On a per-ohm basis, we obtain

p.t/ D P.t/=R D i2.t/ W/ohm

� The average energy and power can be obtain by integratingover the interval jt j � T with T !1

E D limT!1

Z T

�T

i2.t/ dt Joules/ohm

P D limT!1

1

2T

Z T

�T

i2.t/ dt W/ohm

� In system engineering we take the above energy and powerdefinitions, and extend them to an arbitrary signal x.t/, pos-sibly complex, and define the normalized energy (e.g. 1 ohmsystem) as

E�D lim

T!1

Z T

�T

jx.t/j2 dt D

Z1

�1

jx.t/j2 dt

P�D lim

T!1

1

2T

Z T

�T

jx.t/j2 dt



� Signal Classes:

1. x.t/ is an energy signal if and only if 0 < E <1 so thatP D 0

2. x.t/ is a power signal if and only if 0 < P < 1 whichimplies that E !1

Example 2.3: Real Exponential

� Consider x.t/ D Ae�˛tu.t/ where ˛ is real

� For ˛ > 0 the energy is given by

E D

Z1

0

�Ae�˛t

�2dt D

A2e�2˛t

�2˛

ˇ̌̌̌1

0

DA2

2˛

� For ˛ D 0 we just have x.t/ D Au.t/ and E !1

� For ˛ < 0 we also have E !1

� In summary, we conclude that x.t/ is an energy signal for ˛ >0

� For ˛ > 0 the power is given by

P D limT!1

1

2T

A2

2˛

�1 � e�˛T

�D 0

� For ˛ D 0 we have

P D limT!1

1

2T� A2T D

A2

2


2.2. SIGNAL CLASSIFICATIONS

� For ˛ < 0 we have P !1

� In summary, we conclude that x.t/ is a power signal for ˛ D 0

Example 2.4: Real Sinusoid

� Consider x.t/ D A cos.!0t C �/; �1 < t <1

� The signal energy is infinite since upon squaring, and integrat-ing over one cycle, T0 D 2�=!0, we obtain

E D limN!1

Z NT0=2

�NT0=2

A2 cos2.!0t C �/ dt

D limN!1

N

Z T0=2

�T0=2

A2 cos2.!0t C �/ dt

D limN!1

NA2

2

Z T0=2

�T0=2

�1C cos.2!0t C 2�/ dt

D limN!1

NA2

2� T0!1

� The signal average power is finite since the above integral isnormalized by 1=.NT0/, i.e.,

P D limN!1

1

NT0�N

A2

2� T0 D

A2

2



2.3 Generalized Fourier Series

The goal of generalized Fourier series is to obtain a representation ofa signal in terms of points in a signal space or abstract vector space.The coordinate vectors in this case are orthonomal functions. Thecomplex exponential Fourier series is a special case.

� Let EA be a vector in a three dimensional vector space

� Let Ea1; Ea2, and Ea3 be linearly independent vectors in the samethree dimensional space, then

c1Ea1 C c2Ea2 C c3Ea3 D 0 .zero vector/

only if the constants c1 D c2 D c3 D 0

� The vectors Ea1; Ea2, and Ea3 also span the three dimensionalspace, that is for any vector EA there exists a set of constantsc1; c2, and c3 such that

EA D c1Ea1 C c2Ea2 C c3Ea3

� The set fEa1; Ea2; Ea3g forms a basis for the three dimensionalspace

� Now let fEa1; Ea2; Ea3g form an orthogonal basis, which impliesthat

Eai � Eaj D .Eai ; Eaj / D hEai ; Eaj i D 0; i ¤ j

which says the basis vectors are mutually orthogonal

� From analytic geometry (and linear algebra), we know that wecan find a representation for EA as

EA D.Ea1 � EA/

jEa1j2C.Ea2 � EA/

jEa2j2C.Ea3 � EA/

jEa3j2


2.3. GENERALIZED FOURIER SERIES

which implies that

EA D

3XiD1

ci Eai

where

ci DEai � EA

jEai j2; i D 1; 2; 3

is the component of EA in the Eai direction

� We now extend the above concepts to a set of orthogonal func-tions f�1.t/; �2.t/; : : : ; �N .t/g defined on to � t � t0 C T ,where the dot product (inner product) associated with the �n’sis

��m.t/; �n.t/

�D

Z t0CT

t0

�m.t/��

n.t/ dt

D cnımn D

(cn; n D m

0; n ¤ m

� The �n’s are thus orthogonal on the interval Œt0; t0 C T �

� Moving forward, let x.t/ be an arbitrary function on Œt0; t0CT �,and consider approximating x.t/ with a linear combination of�n’s, i.e.,

x.t/ ' xa.t/ D

NXnD1

Xn�n.t/; t0 � t � t0 C T;

where a denotes approximation



� A measure of the approximation error is the integral squarederror (ISE) defined as

�N D

ZT

ˇ̌x.t/ � xa.t/

ˇ̌2dt;

whereRT

denotes integration over any T long interval

� To find the Xn’s giving the minimum �N we expand the aboveintegral into three parts (see homework problems)

�N D

ZT

jx.t/j2 dt �

NXnD1

1

cn

ˇ̌̌̌ZT

x.t/��n.t/ dt

ˇ̌̌̌2C

NXnD1

cn

ˇ̌̌̌Xn �

1

cn

ZT

x.t/��n.t/ dt

ˇ̌̌̌2– Note that the first two terms are independent of the Xn’s

and the last term is nonnegative (missing steps are in texthomework problem 2.14)

� We conclude that �N is minimized for each n if each elementof the last term is made zero by setting

Xn D1

cn

ZT

x.t/��n.t/ dt Fourier Coefficient

� This also results in

��N�

min D

ZT

jx.t/j2 dt �

NXnD1

cnjXnj2



� Definition: The set of of �n’s is complete if

limN!1

.�N /min D 0

forRTjx.t/j2 dt <1

– Even if though the ISE is zero when using a completeset of orthonormal functions, there may be isolated pointswhere x.t/ � xa.t/ ¤ 0

� Summary

x.t/ D l.i.m.1XnD1

Xn�n.t/

Xn D1

cn

ZT

x.t/��n.t/ dt

– The notation l.i.m. stands for limit in the mean, which isa mathematical term referring to the fact that ISE is theconvergence criteria

� Parseval’s theorem: A consequence of completeness isZT

jx.t/j2 dt D

1XnD1

cnjXnj2



Example 2.5: A Three Term Expansion

� Approximate the signal x.t/ D cos 2�t on the interval Œ0; 1�using the following basis functions

0.2 0.4 0.6 0.8 1

-1

-0.75

-0.5

-0.25

0.25

0.5

0.75

1

0.2 0.4 0.6 0.8 1

-1

-0.75

-0.5

-0.25

0.25

0.5

0.75

1x(t) φ

1(t)

t t

0.2 0.4 0.6 0.8 1

-1

-0.75

-0.5

-0.25

0.25

0.5

0.75

1

0.2 0.4 0.6 0.8 1

-1

-0.75

-0.5

-0.25

0.25

0.5

0.75

1

φ2(t) φ

3(t)

t t

Signal x.t/ and basis functions �i.t/; i D 1; 2; 3

� To begin with it should be clear that �1.t/; �2.t/, and �3.t/are mutually orthogonal since the integrand associated with theinner product, �i.t/ � ��j .t/ D 0, for i ¤ j; i; j D 1; 2; 3



� Before finding the Xn’s we need to find the cn’s

c1 D

ZT

j�1.t/j2 dt

Z 1=4

0

j1j2 dt D 1=4

c2 D

ZT

j�2.t/j2 dt D 1=2

c3 D

ZT

j�3.t/j2 dt D 1=4

� Now we can compute the Xn’s:

X1 D 4

ZT

x.t/��1 .t/ dt

D 4

Z 1=4

0

cos.2�t/ dt D2

�sin.2�t/

ˇ̌̌1=40D2

�

X2 D 2

Z 3=4

1=4

cos.2�t/ dt D1

�sin.2�t/

ˇ̌̌3=41=4D�2

�

X3 D 4

Z 1

3=4

cos.2�t/ dt D2

�sin.2�t/

ˇ̌̌13=4D2

�

t

x(t)

xa(t)

0.2 0.4 0.6 0.8 1

-1

-0.75

-0.5

-0.25

0.25

0.5

0.75

1

-2/π

2/π

Functional approximation



� The integral-squared error, �N , can be computed as follows:

�N D

ZT

ˇ̌̌̌x.t/ �

3XnD1

Xn�n.t/

ˇ̌̌̌2dt

D

ZT

jx.t/j2 dt �

3XnD1

cnjXnj2

D1

2�1

4

ˇ̌̌̌2

�

ˇ̌̌̌2�1

2

ˇ̌̌̌2

�

ˇ̌̌̌2�1

4

ˇ̌̌̌2

�

ˇ̌̌̌2D1

2�

ˇ̌̌̌2

�

ˇ̌̌̌2D 0:0947

2.4 Fourier Series

When we choose a particular set of basis functions we arrive at themore familiar Fourier series.

2.4.1 Complex Exponential Fourier Series

� A set of �n’s that is complete is

�n.t/ D ejn!0t ; n D 0;˙1;˙2; : : :

over the interval .t0; t0 C T0/ where !0 D 2�=T0 is the periodof the expansion interval


2.4. FOURIER SERIES

proof of orthogonality��m.t/; �n.t/

�D

Z t0CT0

t0

ejm2�tT0 e

�jn2�tT0 dt D

Z t0CT0

t0

ej 2�T0

.m�n/tdt

D

8̂̂<̂:̂R t0CT0t0

dt; m D nR t0CT0t0

�cosŒ2�.m � n/t=T0�

Cj sinŒ2�.m � n/t=T0��dt; m ¤ n

D

(T0; m D n

0; m ¤ n

We also conclude that cn D T0

� Complex exponential Fourier series summary:

x.t/ D

1XnD�1

Xnejn!0t ; t0 � t � t0 C T0

where Xn D1

T0

ZT0

x.t/e�jn!0t

� The Fourier series expansion is unique

Example 2.6: x.t/ D cos2!0t

� If we expand x.t/ into complex exponentials we can immedi-ately determine the Fourier coefficients

x.t/ D1

2C1

2cos 2!0t

D1

2C1

4ej 2!0t C

1

4e�j 2!0t



� The above implies that

Xn D

8̂̂<̂:̂12; n D 014; n D ˙2

0; otherwise

Time Average Operator

� The time average of signal v.t/ is defined as

hv.t/i�D lim

T!1

1

2T

Z T

�T

v.t/ dt

� Note that

hav1.t/C bv2.t/i D ahv1.t/i C bhv2.t/i;

where a and b are arbitrary constants

� If v.t/ is periodic, with period T0, then

hv.t/i D1

T0

ZT0

v.t/ dt

� The Fourier coefficients can be viewed in terms of the timeaverage operator

� Let v.t/ D x.t/e�jn!0t using e�j� D cos � � j sin � , we findthat

Xn D hv.t/i D hx.t/e�jn!0ti

D hx.t/ cosn!0ti � j hx.t/ sinn!0ti


2.4. FOURIER SERIES

2.4.2 Symmetry Properties of the Fourier Coef-ficients

� For x.t/ real, the following coefficient symmetry propertieshold:

1. X�n D X�n2. jXnj D jX�nj

3. †Xn D �†X�n

proof

X�n D

�1

T0

ZT0

x.t/e�jn!0t dt

��D

1

T0

ZT0

x.t/e�j.�n/!0t dt D X�n

since x�.t/ D x.t/

� Waveform symmetry conditions produce special results too

1. If x.�t / D x.t/ (even function), then

Xn D Re˚Xn; i.e., Im

˚XnD 0

2. If x.�t / D �x.t/ (odd function), then

Xn D Im˚Xn; i.e., Re

˚XnD 0

3. If x.t ˙ T0=2/ D �x.t/ (odd half-wave symmetry), then

Xn D 0 for n even



Example 2.7: Odd Half-wave Symmetry Proof

� Consider

Xn D1

T0

Z t0DT0=2

t0

x.t/e�jn!0t dtC1

T0

Z t0CT0

t0CT0=2

x.t 0/e�jn!0t0

dt 0

� In the second integral we change variables by letting t D t 0 �

T0=2

Xn D1

T0

Z t0CT0=2

t0

x.t/e�jn!0t dt

C1

T0

Z tCT0=2

t0

x.t � T0=2/„ ƒ‚ …�x.t/

e�jn!0.tCT0=2/ dt

D

�1 � e�jn!0T0=2

� 1T0

Z t0CT0=2

t0

x.t/e�jn!0t dt

but n!0.T0=2/ D n.2�=T0/.T0=2/ D n� , thus

1 � e�jn� D

(2; n odd

0; n even

� We thus see that the even indexed Fourier coefficients are in-deed zero under odd half-wave symmetry


2.4. FOURIER SERIES

2.4.3 Trigonometric Form

� The complex exponential Fourier series can be arranged as fol-lows

x.t/ D

1XnD�1

Xnejn!0t

D X0 C

1XnD1

�Xne

jn!0t CX�ne�jn!0t

�� For real x.t/, we may know that jX�nj D jXnj and †Xn D�†X�n, so

x.t/ D X0 C

1XnD1

�jXnje

j Œn!0tC†Xn� C jXnje�j Œn!0tC†Xn�

�D X0 C 2

1XnD1

jXnj cos�n!0t C†Xn

�since cos.x/ D .ejx C e�jx/=2

� From the trig identity cos.uC v/ D cosu cos v� sinu sin v, itfollows that

x.t/ D X0 C

1XnD1

An cos.n!0t /C1XnD1

Bn sin.n!0t /

where

An D 2hx.t/ cos.n!0t /iBn D 2hx.t/ sin.n!0t /i



2.4.4 Parseval’s Theorem

� Fourier series analysis are generally used for periodic signals,i.e., x.t/ D x.t C nT0/ for any integer n

� With this in mind, Parseval’s theorem becomes

P D1

T0

ZT0

jx.t/j2 dt D

1XnD�1

jXnj2

D X20 C 2

1XnD1

jXnj2 .W/

Note: A 1 ohm system is assumed

2.4.5 Line Spectra

� Line spectra was briefly reviewed earlier for simple signals

� For any periodic signal having Fourier series representation wecan obtain both single-sided and double-sided line spectra

� The double-sided magnitude and phase line spectra is mosteasily obtained form the complex exponential Fourier series,while the single-sided magnitude and phase line spectra can beobtained from the trigonometric form

Double-sidedmag. and phase

()

1XnD�1

Xnej 2�.nf0/t

Single-sidedmag. and phase

() X0 C 2

1XnD1

jXnj cosŒ2�.nf0/t C†Xn�


2.4. FOURIER SERIES

– For the double-sided simply plot as lines jXnj and †Xnversus nf0 for n D 0;˙1;˙2; : : :

– For the single-sided plot jX0j and †X0 as a special casefor n D 0 at nf0 D 0 and then plot 2jXnj and †X0 versusnf0 for n D 1; 2; : : :

Example 2.8: Cosine Squared

� Consider

x.t/ D A cos2.2�f0t C �/ DA

2CA

2cos

�2�.2f0/t C 2�1

�DA

2CA

4ej 2�1ej 2�.2f0/t C

A

4e�j 2�1e�j 2�.2f0/t

Double-Sided

Single-Sided

-2f0

-2f0

2f0

2f0

2f0

2f0

f

f f

f



Example 2.9: Pulse Train

t

A

x(t)

0 τ T0T

0 + τ-T

0-2T

0

. . .. . .

Periodic pulse train

� The pulse train signal is mathematically described by

x.t/ D

1XnD�1

A…

�t � nT0 � �=2

�

�

� The Fourier coefficients are

Xn D1

T0

Z �

0

Ae�j 2�.nf0/t dt DA

T0�e�j 2�.nf0/t

�j 2�.nf0/

ˇ̌̌�0

DA

T0�1 � e�j 2�.nf0/�

j 2�.nf0/

DA�

T0�ej�.nf0/� � e�j�.nf0/�

.2j /�.nf0/�� e�j�.nf0/�

DA�

T0�

sinŒ�.nf0/��Œ�.nf0/��

� e�j�.nf0/�

� To simplify further we define

sinc.x/ �D

sin.�x/�x


2.4. FOURIER SERIES

� Finally,

Xn DA�

T0sinc.nf0�/e�j�.nf0/� ; n D 0;˙1;˙2; : : :

� To plot the line spectra we need to find jXnj and †Xn

jXnj DA�

T0jsincŒ.nfo/��j

†Xn D

8̂̂<̂:̂��.nf0/�; sinc.nfo�/ > 0

��.nf0/� C �; nf0 > 0 and sinc.nf0�/ < 0

��.nf0/� � �; nf0 < 0 and sinc.nf0�/ < 0

� Plot some double-sided line spectra example using MATLAB

� First we create a helper function that takes as input a vector offrequency values nf0 and the coefficients Xn

function Line_Spectra(fk,Xk,mode)% Line_Spectra(fk,Xk,mode1) (file Line_Spectra.m)%% Plot Double-Sided Line Spectra%----------------------------------------------------% fk = vector of real sinusoid frequencies% Xk = magnitude and phase at each frequency in fk% mode = ’mag’ or ’phase’ plot%% % Mark Wickert, January 2007

switch lower(mode) % not case sensitivecase ’mag’,’magnitude’ % two choices work

stem(fk,abs(Xk),’filled’);gridaxis([-1.05*max(fk) 1.05*max(fk) 0 1.05*max(abs(Xk))])ylabel(’Magnitude’)xlabel(’Frequency (Hz)’)

case ’phase’stem(fk,angle(Xk),’filled’);gridaxis([-1.05*max(fk) 1.05*max(fk), ...-1.1*max(abs(angle(Xk))) 1.1*max(abs(angle(Xk)))])

ylabel(’Phase (rad)’)



xlabel(’Frequency (Hz)’)otherwise

error(’mode must be mag or phase’)end

� As a specific example enter the following at the MATLAB com-mand prompt

>> n = -25:25;>> tau = 0.125; f0 = 1; A = 1;>> Xn = A*tau*f0*sinc(n*f0*tau).*exp(-j*pi*n*f0*tau);>> subplot(211)>> Line_Spectra(n*f0,Xn,’mag’)>> subplot(212)>> Line_Spectra(n*f0,Xn,’phase’)

−25 −20 −15 −10 −5 0 5 10 15 20 250

0.05

0.1

Mag

nitu

de

Frequency (Hz)

−25 −20 −15 −10 −5 0 5 10 15 20 25

−2

0

2

Pha

se (

rad)

Frequency (Hz)

1/τ = 8

f0 = 1, τ = 0.125Aτf

0 = 0.125


2.4. FOURIER SERIES

2.4.6 Numerical Calculation of Xn� Here we consider a purely numerical calculation of the Xk co-

efficients from a single period waveform description of x.t/

� In particular, we will use MATLAB’s fast Fourier transform(FFT) function to carry out the numerical integration

� By definition

Xk D1

T0

ZT0

x.t/e�j 2�kf0t dt; k D 0;˙1;˙2; : : :

� A simple rectangular integration approximation to the aboveintegral is

Xk '1

T0

N�1XnD0

x.nT /e�jk2�.nf0/T0=N �T0

N; k D 0;˙1;˙2; : : :

where N is the number of points used to partition the timeinterval Œ0; T0� and T D T0=N is the time step

� Using the fact that 2�f0T0 D 2� , we can write that

Xk '1

N

N�1XnD0

x.nT /e�j 2�kn

N ; k D 0;˙1;˙2; : : :

� Note that the above must be evaluated for each Fourier coeffi-cient of interest

� Also note that the accuracy of the Xk values depends on thevalue of N



– For k small and x.t/ smooth in the sense that the harmon-ics rolloff quickly,N on the order of 100 may be adequate

– For k moderate, say 5–50, N will have to become in-creasingly larger to maintain precision in the numericalintegral

Calculation Using the FFT

� The FFT is a powerful digital signal processing (DSP) func-tion, which is a computationally efficient version of the dis-crete Fourier transfrom (DFT)

� For the purposes of the problem at hand, suffice it to say thatthe FFT is just an efficient algorithm for computing

XŒk� D

N�1XnD0

xŒn�e�j 2�kn=N ; k D 0; 1; 2; : : : ; N � 1

� If we let xŒn� D x.nT /, then it should be clear that

Xk '1

NXŒk�; k D 0; 1; : : : ;

N

2

� To obtain Xk for k < 0 note that

X�k '1

NXŒ�k� D

1

N

N�1XnD0

x.nT /e�j 2�.�k/n

N

D1

N

N�1XnD0

x.nT /e�j 2�.N�k/n

N D XŒN � k�

since e�j 2�Nn=N D e�j 2�n D 1


2.4. FOURIER SERIES

� In summary

Xk '

(XŒk�=N; 0 � k � N=2

XŒN � k�=N; �N=2 � k < 0

� To use the MATLAB function fft() to obtain theXk we simplylet

X D fft(x)

where xD fx.t/ W t D 0; T0=N; 2T0=N; : : : ; T0.N � 1/=N g

� Remember in MATLAB that XŒ0� is really found in X[1], etc.

Example 2.10: Finite Rise/Fall-Time PulseTrain

t

x(t)1

1/2

0 tr T

0τ

τ

τ + tr

Pulse width = τRise and fall time = t

r

Pulse train with finite rise and fall time edges

� Shown above is one period of a finite rise and fall time pulsetrain

� We will numerically compute the Fourier series coefficients ofthis signal using the FFT

� The MATLAB function trap_pulse was written to generateone period of the signal using N samples



function [xp,t] = trap_pulse(N,tau,tr)% xp = trap_pulse(N,tau,tr)%% Mark Wickert, January 2007

n = 0:N-1;t = n/N;xp = zeros(size(t));% Assume tr and tf are equalT1 = tau + tr;% Create one period of the trapezoidal pulse waveformfor k=1:N

if t(k) <= trxp(k) = t(k)/tr;

elseif (t(k) > tr & t(k) <= tau)xp(k) = 1;

elseif (t(k) > tau & t(k) < T1)xp(k) = -t(k)/tr+ 1 + tau/tr;

elsexp(k) = 0;

endend

� We now plot the double-sided line spectra for � D 1=8 and twovalues of rise-time tr

>> % tau = 1/8, tr = 1/20>> N = 1024;>> [xp,t] = trap_pulse(N,1/8,1/20);>> Xp = fft(xp);>> subplot(211)>> plot(t,xp)>> grid>> ylabel(’x(t)’)>> xlabel(’Time (s)’)>> subplot(212)>> Xp_shift = fftshift(Xp)/N;>> f = -N/2:N/2-1;>> Line_Spectra(f,Xp_shift,’mag’)>> axis([-25 25 0 .15])>> print -tiff -depsc line_spec2.eps>> % tau = 1/8, tr = 1/10>> xp = trap_pulse(N,1/8,1/10);>> Xp = fft(xp);


2.4. FOURIER SERIES

>> Xp_shift = fftshift(Xp)/N;>> f = N/2:N/2-1;>> subplot(211)>> plot(t,xp)>> grid>> ylabel(’x(t)’)>> xlabel(’Time (s)’)>> subplot(212)>> Line_Spectra(f,Xp_shift,’mag’)>> axis([-25 25 0 .15])>> print -tiff -depsc line_spec3.eps

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10

0.2

0.4

0.6

0.8

1

x(t)

Time (s)

−25 −20 −15 −10 −5 0 5 10 15 20 250

0.05

0.1

0.15

Mag

nitu

de

Frequency (Hz)

1/τ = 1/8

1/8

1/20

Sidelobes smaller than ideal pulse train which has zero rise time

f0 = 1, τ = 0.125, t

r = 1/20

Signal x.t/ and line spectrum for � D 1=8 and tr D 1=20



0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10

0.2

0.4

0.6

0.8

1x(

t)

Time (s)

−25 −20 −15 −10 −5 0 5 10 15 20 250

0.05

0.1

0.15

Mag

nitu

de

Frequency (Hz)

1/τ = 1/8

1/8

1/10

Sidelobes smaller than with t

r = 1/20

case

f0 = 1, τ = 0.125, t

r = 1/10

Signal x.t/ and line spectrum for � D 1=8 and tr D 1=10


2.4. FOURIER SERIES

2.4.7 Other Fourier Series Properties

� Given x.t/ has Fourier series (FS) coefficients Xn, if

y.t/ D AC Bx.t/

it follows that

Yn D

(AC BX0; n D 0

BXn; n ¤ 0

proof:

Yn D hy.t/e�j 2�.nf0/ti

D Ahe�j 2�.nf0/ti C Bhx.t/e�j 2�.nf0/ti

D A

�1; n D 0

0; n ¤ 0

�C BXn

QED

� Likewise ify.t/ D x.t � t0/

it follows thatYn D Xne

�j 2�.nf0/t0

proof:Yn D hx.t � t0/e

�j 2�.nf0/ti

Let � D t � t0 which implies also that t D �C t0, so

Yn D hx.�/e�j 2�.nf0/.�Ct0/i

D hx.�/e�j 2�.nf0/�ie�j 2�.nf0/t0

D Xne�j 2�.nf0/t0

QED



2.5 Fourier Transform

� The Fourier series provides a frequency domain representationof a periodic signal via the Fourier coefficients and line spec-trum

� The next step is to consider the frequency domain representa-tion of aperiodic signals using the Fourier transform

� Ultimately we will be able to include periodic signals withinthe framework of the Fourier transform, using the concept oftransform in the limit

� The text establishes the Fourier transform by considering alimiting case of the expression for the Fourier series coefficientXn as T0!1

� The Fourier transform (FT) and inverse Fourier transfrom (IFT)is defined as

X.f / D

Z1

�1

x.t/e�j 2�f t dt (FT)

x.t/ D

Z1

�1

X.f /ej 2�f t df (IFT)

� Sufficient conditions for the existence of the Fourier transformare

1.R1

�1jx.t/j dt <1

2. Discontinuities in x.t/ be finite

3. An alternate sufficient condition is thatR1

�1jx.t/j2 dt <

1, which implies that x.t/ is an energy signal


2.5. FOURIER TRANSFORM

2.5.1 Amplitude and Phase Spectra

� FT properties are very similar to those obtained for the Fouriercoefficients of periodic signals

� The FT, X.f / D Ffx.t/g, is a complex function of f

X.f / D jX.f /jej�.f / D jX.f /jej†X.f /

D RefX.f /g C j ImfX.f /g

� The magnitude jX.f /j is referred to as the amplitude spectrum

� The the angle †X.f / is referred to as the phase spectrum

� Note that

RefX.f /g DZ1

�1

x.t/ cos 2�f t dt

ImfX.f /g D �Z1

�1

x.t/ sin 2�f t dt

2.5.2 Symmetry Properties

� If x.t/ is real it follows that

X.�f / D

Z1

�1

x.t/e�j 2�.�f /t dt

D

�Z1

�1

x.t/e�j 2�f t dt

��D X�.f /

thus

jX.�f /j D jX.f /j (even in frequency)†X.�f / D �†X.f / (odd in frequency)



� Additionally,

1. For x.�t / D x.t/ (even function), ImfX.f /g D 0

2. For x.�t / D �x.t/ (odd function), RefX.f /g D 0

2.5.3 Energy Spectral Density

� From the definition of signal energy,

E D

Z1

�1

jx.t/j2 dt

D

Z1

�1

x�.t/

�Z1

�1

X.f /ej 2�f t df

�dt

D

Z1

�1

X.f /

�Z1

�1

x�.t/ej 2�f t dt

�df

butZ1

�1

x�.t/ej 2�f t dt D

�Z1

�1


��D X�.f /

� Finally,

E D

Z1

�1

jx.t/j2 dt D

Z1

�1

jX.f /j2 df

which is known as Rayleigh’s Energy Theorem

� Are the units consistent?

– Suppose x.t/ has units of volts

– jX.f /j2 has units of (volts-sec)2



– In a 1 ohm system jX.f /j2 has units of Watts-sec/Hz =Joules/Hz

� The energy spectral density is defined as

G.f /�D jX.f /j2 Joules/Hz

� It then follows that

E D

Z1

�1

G.f / df

Example 2.11: Rectangular Pulse

� Consider

x.t/ D A…

�t � t0

�

�� FT is

X.f / D A

Z t0C�=2

t0��=2

e�j 2�f t dt

D A �e�j 2�f t

�j 2�f

ˇ̌̌̌ˇt0C�=2

t0��=2

D A� �

�ej�f � � e�j�f �

.j 2/�f �

�� e�j 2�f t0

D A�sinc.f �/e�j 2�f t0

A…

�t � t0

�

�F ! A�sinc.f �/e�j 2�f t0



� Plot jX.f /j, †X.f /, and G.f /

? 3 ? 2 ? 1 1 2 3

0.2

0.4

0.6

0.8

1

? 3 ? 2 ? 1 1 2 3

? 3

? 2

? 1

1

2

3

? 3 ? 2 ? 1 1 2 3

0.2

0.4

0.6

0.8

1

f

f

f

0-1/τ 1/τ 2/τ-2/τ

0-1/τ 1/τ 2/τ-2/τ

-1/τ 1/τ 2/τ-2/τ

|X(f)|

G(f) = |X(f)|2

X(f)AmplitudeSpectrum

EnergySpectralDensity

PhaseSpectrum

Aτ

(Aτ)2

π

π/2

−π/2

−πt0 = τ/2 slope = -πfτ/2

Rectangular pulse spectra

2.5.4 Transform Theorems

� Be familiar with the FT theorems found in the table of Ap-pendix G.6 of the text

Superposition Theorem

a1x1.t/C a2x2.t/F ! a1X1.f /C a2X2.f /

proof:



Time Delay Theorem

x.t � t0/F ! X.f /e�j 2�f t0

proof:

Frequency Translation Theorem

� In communications systems the frequency translation and mod-ulation theorems are particularly important

x.t/ej 2�f0tF ! X.f � f0/

proof: Note thatZ1

�1

x.t/ej 2�f0te�j 2�f t dt D

Z1

�1

x.t/e�j 2�.f �f0/t dt

soF˚x.t/ej 2�f0t

D X.f � f0/

QED

Modulation Theorem

� The modulation theorem is an extension of the frequency trans-lation theorm

x.t/ cos.2�f0t /F !

1

2X.f � f0/C

1

2X.f C f0/



proof: Begin by expanding

cos.2�f0t / D1

2ej 2�f0t C

1

2e�j 2�f0t

Then apply the frequency translation theorem to each term sep-arately

x(t) y(t)

cos(2πf0t)

X(f)Y(f)

f f

A/2

A

f0

-f0

00

signalmultiplier

A simple modulator

Duality Theorem

� Note that

FfX.t/g DZ1

�1

X.t/e�j 2�f t dt D

Z1

�1

X.t/ej 2�.�f /t dt

which implies that

X.t/F ! x.�f /



Example 2.12: Rectangular Spectrum

f-W W0

1X(f)

� Using duality on the above we have

X.t/ D …

�t

2W

�F ! 2W sinc.2Wf / D x.�f /

� Since sinc( ) is an even function (sinc.x/ D sinc.�x/), it fol-lows that

2W sinc.2W t/F ! …

�f

2W

�

Differentiation Theorem

� The general result isd nx.t/

dtnF ! .j 2�f /n X.f /

proof: For n D 1we start with the integration by parts formula,Rudv D uv

ˇ̌̌�Rv du, and apply it to

F�dx

dt

�D

Z1

�1

dx

dte�j 2�f t dt

D x.t/e�j 2�f tˇ̌̌1

�1„ ƒ‚ …0

Cj 2�f

Z1

�1

x.t/e�j 2�f t dt„ ƒ‚ …X.f /



alternate — From Leibnitz’s rule for differentiation of inte-grals,

d

dt

Z1

�1

F.f; t/ df D

Z1

�1

@F.f; t/

@fdf

sodx.t/

dtDd

dt

Z1

�1

X.f /ej 2�f t df

D

Z1

�1

X.f /@ej 2�f t

@tdf

D

Z1

�1

j 2�fX.f /ej 2�f t df

) dx=dtF ! j 2�fX.f / QED

Example 2.13: FT of Triangle Pulse

τ−τ 0t

1

� Note that

ττ−τ −τ

t t

1/τ

1/τ

-2/τ-1/τ



� Using the differentiation theorem for n D 2 we have that

F�ƒ

�t

�

��D

1

.j 2�f /2Fn1�ı.t C �/ �

2

�ı.t/C

1

�ı.t � �/

oD

1�ej 2�f � � 2

�C

1�e�j 2�f �

.j 2�f /2

D2 cos.2�f �/ � 2��.2�f /2

D �4 sin2.�f �/4.�f �/2

D �sinc2.f �/

ƒ

�t

�

�F ! �sinc2.f �/

Convolution and Convolution Theorem

� Before discussing the convolution theorem we need to reviewconvolution

� The convolution of two signals x1.t/ and x2.t/ is defined as

x.t/ D x1.t/ � x2.t/ D

Z1

�1

x1.�/x2.t � �/ d�

D x2.t/ � x1.t/ D

Z1

�1

x2.�/x1.t � �/ d�

� A special convolution case is ı.t � t0/

ı.t � t0/ � x.t/ D

Z1

�1

ı.� � t0/x.t � �/ d�

D x.t � �/ˇ̌�Dt0D x.t � t0/



Example 2.14: Rectangular Pulse Convolution

� Let x1.t/ D x2.t/ D ….t=�/

� To evaluate the convolution integral we need to consider theintegrand by sketching of x1.�/ and x2.t ��/ on the � axis fordifferent values of t

� For this example four cases are needed for t to cover the entiretime axis t 2 .�1;1/

� Case 1: When t < � we have no overlap so the integrand iszero and x.t/ is zero

Qt t + Y/2t - Y/2 0 Y/2�Y/2

x2(t - Q) x1(Q)No overlap for t + Y/2 < –Y/2 or t < –Y

� Case 2: When �� < t < 0 we have overlap and

x.t/ D

Z1

�1

x1.�/x2.t � �/ d�

D

Z tC�=2

��=2

d� D �ˇ̌̌tC�=2��=2

D t C �=2C �=2 D � C t

Overlap begins when t + τ/2 = -τ/2 or t = -τ

λ

t + τ/20 τ/2−τ/2

x2(t - λ) x

1(λ)



� Case 3: For 0 < t < � the leading edge of x2.t � �/ is to theright of x1.�/, but the trailing edge of the pulse is still over-lapped

x.t/ D

Z �=2

t��=2

d� D �=2 � t C �=2 D � � t

λt + τ/2

t - τ/20 τ/2−τ/2

x2(t - λ)x

1(λ)

Overlap lasts until t = τ

� Case 4: For t > � we have no overlap, and like case 1, theresult is

x.t/ D 0

λt + τ/2t - τ/20 τ/2−τ/2

x2(t - λ)x

1(λ)

No overlap for t > τ

� Collecting the results

x.t/ D

8̂̂̂̂<̂ˆ̂̂:0; t < ��

� C t; �� t < 0

� � t; 0 � t < �

0; t � �

D

(� � jt j; jt j � �

0; otherwise



� Final summary,

…

�t

�

��…

�t

�

�D �ƒ

�t

�

�

� Convolution Theorem: We now consider x1.t/�x2.t/ in termsof the FTZ

1

�1

x1.�/x2.t � �/ d�

D

Z1

�1

x1.�/

�Z1

�1

X2.f /ej 2�f .t��/ df

�d�

D

Z1

�1

X2.f /

�Z1

�1

x1.�/e�j 2�f � d�

�ej 2�f t df

D

Z1

�1

X1.f /X2.f /ej 2�f t df

which implies that

x1.t/ � x2.t/F ! X1.f /X2.f /

Example 2.15: Revisit ….t=�/ �….t=�/

� Knowing that….t=�/�….t=�/ D �ƒ.t=�/ in the time domain,we can follow-up in the frequency domain by writing

F˚….t=�/

� F˚….t=�/

D��sinc.f �/

�2� We have also established the transform pair

�ƒ

�t

�

�F ! �2sinc2.f �/ D �

��sinc2.f �/



or

ƒ

�t

�

�F ! �sinc2.f �/

Multiplication Theorem

� Having already established the convolution theorem, it followsfrom the duality theorem or direct evaluation, that

x1.t/ � x2.t/F ! X1.f / �X2.f /

2.5.5 Fourier Transforms in the Limit

� Thus far we have considered two classes of signals

1. Periodic power signals which are described by line spec-tra

2. Non-periodic (aperiodic) energy signals which are describedby continuous spectra via the FT

� We would like to have a unifying approach to spectral analysis

� To do so we must allow impulses in the frequency domain byusing limiting operations on conventional FT pairs, known asFourier transforms-in-the-limit

– Note: The corresponding time functions have infinite en-ergy, which implies that the concept of energy spectraldensity will not apply for these signals (we will introducethe concept of power spectral density for these signals)



Example 2.16: A Constant Signal

� Let x.t/ D A for �1 < t <1

� We can writex.t/ D lim

T!1A….t=T /

� Note thatF˚A….t=T /

D AT sinc.f T /

� Using the transform-in-the-limit approach we write

Ffx.t/g D limT!1

AT sinc.f T /

? 3 ? 2 ? 1 1 2 3? 0.2

0.2

0.4

0.6

0.8

1

? 3 ? 2 ? 1 1 2 3? 0.2

0.2

0.4

0.6

0.8

1AT1

AT2

T2 >> T

1

f f

Increasing T in AT sinc.f T /

� Note that since x.t/ has no time variation it seems reasonablethat the spectral content ought to be confined to f D 0

� Also note that it can be shown thatZ1

�1

AT sinc.f T / df D A; 8 T

� Thus we have established that

AF ! Aı.f /



� As a further check

F�1˚Aı.f /

D

Z1

�1

Aı.f /ej 2�ff t df D Aej 2�f tˇ̌̌fD0D A

� As a result of the above example, we can obtain several moreFT-in-the-limit pairs

Aej 2�f0tF ! Aı.f � f0/

A cos.2�f0t C �/F !

A

2

�ej�ı.f � f0/C e

�j�ı.f C f0/�

Aı.t � t0/F ! Ae�j 2�f t0

� Reciprocal Spreading Property: Compare

Aı.t/F ! A and A

F ! Aı.f /

A constant signal of infinite duration has zero spectral width,while an impulse in time has zero duration and infinite spectralwidth

2.5.6 Fourier Transforms of Periodic Signals

� For an arbitrary periodic signal with Fourier series

x.t/ D

1XnD�1

Xnej 2�nf0t



we can write

X.f / D F"1X

nD�1

Xnej 2�nf0t

#

D

1XnD�1

XnFnej 2�nf0t

oD

1XnD�1

Xnı.f � nf0/

using superposition and FfAej 2�f0tg D Aı.f � f0/

� What is the difference between line spectra and continuousspectra? none!

� Mathematically,

LineSpectra

Convert to time domain

Convert to time domain

Sum phasors

Integrate impulses to get phasors via the inverse FT

ContinuousSpectra

� The Fourier series coefficients need to be known before the FTspectra can be obtained

� A technique that obtained the FT directly will be discussedlater



Example 2.17: Ideal Sampling Waveform

� When we discuss sampling theory it will be useful to have theFT of the periodic impulse train signal

ys.t/ D

1XmD�1

ı.t �mTs/

where Ts is the sample spacing or period

� Since this signal is periodic, it must have a Fourier series rep-resentation too

� In particular

Yn D1

Ts

ZTs

ı.t/e�j 2�.nfs/t dt D1

TsD fs; any n

where fs is the sampling rate in Hz

� The FT of ys.t/ is given by

Ys.f / D fs

1XnD�1

F˚ej 2�nf0/t

D fs

1XnD�1

ı.f � nfs/

� Summary,

1XmD�1

ı.t �mTs/F ! fs

1XnD�1

ı.f � nfs/



. . . . . .

t

f

ys(t)

Ys(f)

0

0

Ts

fs 4f

s

4Ts

-Ts

-fs

. . . . . .fs

1

An impulse train in times is an impulse train in frequency

Example 2.18: Convolve Step and Exponential

� Find y.t/ D Au.t/ � e�˛tu.t/, ˛ > 0

� For t � 0 there is no overlap so Y.t/ D 0

λ0t

No overlap



� For t > 0 there is always overlap

y.t/ D

Z t

0

A � e�˛.t��/ d�

D Ae�˛t �e˛�

˛

ˇ̌̌t0

D Ae�˛t �e˛t � 1

˛

λ0 t

For t > 0 there is always overlap

� Summary,

y.t/ DA

˛

�1 � e�˛t

�u.t/

t

y(t)

A/α

0



Direct Approach for the FT of a Periodic Signal

� The FT of a periodic signal can be found directly by expandingx.t/ as follows

x.t/ D

"1X

mD�1

ı.t �mTs/

#� p.t/ D

1XmD�1

p.t �mTs/

where p.t/ represents one period of x.t/, having period Ts

� From the convolution theorem

X.f / D F(1X

mD�1

ı.t �mTs/

)� P.f /

D fsP.f /

1XnD�1

ı.f � nfs/

D fs

1XnD�1

P.nfs/ı.f � nfs/

where P.f / D Ffp.t/g

� The FT transform pair just established is

1XmD�1

p.t �mTs/F !

1XnD�1

fsP.nfs/ı.f � nfs/



Example 2.19: p.t/ D ….t=2/C….t=4/, T0 D 10

tT

0 = 100 1-1-2 2

. . .. . .

x(t)

1

2

Stacked pulses periodic signal

� We begin by finding P.f / using Ff….t=�/g D �sinc.f �/

P.f / D 2sinc.2f /C 4sinc.4f /

� Plugging into the FT pair derived above with nfs D n=10,

X.f / D1

10

1XnD�1

�2sinc

�n5

�C 4sinc

�2n

5

��ı�f �

n

10

�

2.5.7 Poisson Sum Formula

� The Poisson sum formula from mathematics can be derivedusing the FT pair

e�j 2�.nfs/tF ! ı.f � nfs/

by writing

F�1(1X

nD�1

fsP.nfs/ı.f � nfs/

)D fs

1XnD�1

P.nfs/ej 2�.nfs/t



� From the earlier developed FT of periodic signals pair, weknow that the left side of the above is also equal to

1XmD�1

p.t �mTs/alsoD fs

1XnD�1

P.nfs/ej 2�.nfs/t

� We can finally relate this back to the Fourier series coefficients,i.e.,

Xn D fsP.nfs/

2.6 Power Spectral Density and Corre-lation

� For energy signals we have the energy spectral density, G.f /,defined such that

E D

Z1

�1

G.f / df

� For power signals we can define the power spectral density(PSD), S.f / of x.t/ such that

P D

Z1

�1

S.f / df D hjx.t/j2i

– Note: S.f / is real, even and nonnegative

– If x.t/ is periodic S.f / will consist of impulses at theharmonic locations

� For x.t/ D A cos.!0t C �/, intuitively,

S.f / D1

4A2ı.f � f0/C

1

4A2ı.f C f0/


2.6. POWER SPECTRAL DENSITY AND CORRELATION

sinceRS.f / df D A2=2 as expected (power on a per ohm

basis)

� To derive a general formula for the PSD we first need to con-sider the autocorrelation function

2.6.1 The Time Average Autocorrelation Func-tion

� Let �.�/ be the autocorrelation function of an energy signal

�.�/ D F�1˚G.f /

D F�1

˚X.f /X�.f /

D F�1

˚X.f /

� F�1

˚X�.f /

but x.�t /

F ! X�.f / for x.t/ real, so

�.�/ D x.t/ � x.�t / D

Z1

�1

x.t/x.t C �/ dt

or

�.�/ D limT!1

Z T

�T

x.t/x.t C �/ dt

� Observe thatG.f / D F

˚�.�/

� The autocorrelation function (ACF) gives a measure of the

similarity of a signal at time t to that at time t C � ; the co-herence between the signal and the delayed signal



x(t)

X(f)G(f) = |X(f)|2

φ(τ) =

Energy spectral density and signal relationships

2.6.2 Power Signal Case

For power signals we define the autocorrelation function as

Rx.�/ D hx.t/x.t C �/i

D limT!1

1

2T

Z T

�T

x.t/x.t C �/ dt

if periodicD

1

T0

ZT0

x.t/x.t C �/ dt

� Note that

Rx.0/ D hjx.t/j2i D

Z1

�1

Sx.f / df

and since for energy signals �.�/F ! G.f /, a reasonable

assumption is that

Rx.�/F ! Sx.f /



� A formal statement of this is the Wiener-Kinchine theorem (aproof is given in text Chapter 6)

Sx.f / D

Z1

�1

Rx.�/e�j 2�f � d�

x(t) Rx(τ) S

x(f)

Power spectral density and signal relationships

2.6.3 Properties of R.�/

� The following properties hold for the autocorrelation function

1. R.0/ D hjx.t/j2i � jR.�/j for all values of �

2. R.��/ D hx.t/x.t � �/i D R.�/ ) an even function

3. limj� j!1R.�/ D hx.t/i2 if x.t/ is not periodic

4. If x.t/ is periodic, with period T0, thenR.�/ D R.�CT0/

5. FfR.�/g D S.f / � 0 for all values of f

� The power spectrum and autocorrelation function are frequentlyused for systems analysis with random signals



Example 2.20: Single Sinusoid

� Consider the signal x.t/ D A cos.2�f0t C �/, for all t

Rx.�/ D1

T0

Z T0

0

A2 cos.2�f0t C �/ cos.2�.t C �/C �/ dt

DA2

2T0

ZT0

�cos.2�f0�/C cos.2�.2f0/t C 2�f0� C 2�/

�dt

DA2

2cos.2�f0�/

� Note that

F˚Rx.�/

D Sx.f / D

A2

4

�ı.f � f0/C ı.f C f0/

�

More Autocorrelation Function Properties

� Suppose that x.t/ has autocorrelation function Rx.�/

� Let y.t/ D AC x.t/, A D constant

Ry.�/ D hŒAC x.t/�ŒAC x.t C �/�i

D hA2i C hAx.t C �/i C hAx.t/i C hx.t/x.t C �/i

D A2 C 2Ahx.t/i„ ƒ‚ …const. terms

CRx.�/

� Let z.t/ D x.t � t0/

Rz.�/ D hz.t/z.t C �/i D hx.t � t0/x.t � t0 C �/i

D hx.�/x.�C �/i; with � D t � t0D Rx.�/



� The last result shows us that the autocorrelation function isblind to time offsets

Example 2.21: Sum of Two Sinusoids

� Consider the sum of two sinusoids

y.t/ D x1.t/C x2.t/

where x1.t/ D A1 cos.2�f1tC�1/ and x2.t/ D A2 cos.2�f2tC�2/ and we assume that f1 ¤ f2

� Using the definition

Ry.�/ D hŒx1.t/C x2.t/�Œx1.t C �/x2.t C �/�i

D hx1.t/x1.t C �/i C hx2.t/x2.t C �/i

C hx1.t/x2.t C �/i C hx2.t/x1.t C �/i

� The last two terms are zero since hcos..!1˙!2/t/i D 0 whenf1 ¤ f2 (why?), hence

Ry.�/ D Rx1.�/CRx2.�/; for f1 ¤ f2

DA212

cos.2�f1�/CA222

cos.2�f2�/

Example 2.22: PN Sequences

� In the testing and evaluation of digital communication systemsa source of known digital data (i.e., ‘1’s and ‘0’s) is required(see text Chapter 9 p. 507–510)



� A maximal length sequence generator or pseudo-noise (PN)code is often used for this purpose

� Practical implementation of a PN code generator can be ac-complished using an N -stage shift register with appropriateexclusive-or feedback connections

� The sequence length or period of the resulting PN code isM D2N � 1 bits long

CD

1Q

1

CD

2Q

2

CD

3Q

3

M = 23 - 1 = 7

one period = NT

t

x(t)

x(t)

ClockPeriod = T

+A

-A

Three stage PN (m-sequence) generator using logic circuits

� PN sequences have quite a number of properties, one being thatthe time average autocorrelation function is of the form shownbelow



τ

Rx(τ)

T-T

MT

MT

. . .. . .

A2

-A2/M

PN sequence autocorrelation function

� The calculation of the power spectral density will be left as ahomework problem (a specific example is text Example 2.20)

– Hint: To find Sx.f / D FfRx.�/g we useXn

p.t � nTs/F ! fs

Xn

P.nfs/ı.f � nfs/

where Ts DMT

– One period of Rx.�/ is a triangle pulse with a level shift

� Suppose the logic levels are switched from˙A to positive lev-els of say v1 to v2

– Using the additional autocorrelation function propertiesthis can be done

– You need to know that a PN sequence contains one more‘1’ than ‘0’

� MATLAB code for generating PN sequences from 2 to 12 stagesis given below

function c = m_seq(m)%function c = m_seq(m)%% Generate an m-sequence vector using an all-ones initialization



%% Mark Wickert, April 2005

sr = ones(1,m);

Q = 2^m - 1;c = zeros(1,Q);

switch mcase 2

taps = [1 1 1];case 3

taps = [1 0 1 1];case 4

taps = [1 0 0 1 1];case 5

taps = [1 0 0 1 0 1];case 6

taps = [1 0 0 0 0 1 1];case 7

taps = [1 0 0 0 1 0 0 1];case 8

taps = [1 0 0 0 1 1 1 0 1];case 9

taps = [1 0 0 0 0 1 0 0 0 1];case 10

taps = [1 0 0 0 0 0 0 1 0 0 1];case 11

taps = [1 0 0 0 0 0 0 0 0 1 0 1];case 12

taps = [1 0 0 0 0 0 1 0 1 0 0 1 1];otherwise

disp(’Invalid length specified’)end

for n=1:Q,tap_xor = 0;c(n) = sr(m);for k=2:m,

if taps(k) == 1,tap_xor = xor(tap_xor,xor(sr(m),sr(m-k+1)));

endendsr(2:end) = sr(1:end-1);sr(1) = tap_xor;

end



R.�/, S.f /, and Fourier Series

� For a periodic power signal, x.t/, we can write

x.t/ D

1XnD�1

Xnej 2�.nf0/t

� There is an interesting linkage between the Fourier series rep-resentation of a signal, the power spectrum, and then back tothe autocorrelation function

� Using the orthogonality properties of the Fourier series expan-sion we can write

R.�/ D

* 1X

nD�1

Xnej 2�.nf0/t

! 1X

mD�1

Xmej 2�.mf0/.tC�/

!�+

D

1XnD�1

1XmD�1

XnX�

m

˝ej 2�.nf0t /e�j 2�.mf0/.tC�/

˛„ ƒ‚ …n¤m terms are zero, why?

D

1XnD�1

jXnj2˝ej 2�.nf0/te�j 2�.nf0/.tC�/

˛D

1XnD�1

jXnj2ej 2�.nf0/�

� The power spectral density can be obtained by Fourier trans-forming both sides of the above

S.f / D

1XnD�1

jXnj2ı.f � nf0/



2.7 Linear Time Invariant (LTI) Systems

x(t) y(t) =

operator

Linear system block diagram

Definition

� Linearity (superposition) holds, that is for input ˛1x1.t/C˛2x2.t/,˛1 and ˛2 constants,

y.t/ D H�˛1x1.t/C ˛2x2.t/

�D ˛1H

�x1.t/

�C ˛2H

�x2.t/

�D ˛1y1.t/C ˛2y2.t/

� A system is time invariant (fixed) if for y.t/ D HŒx.t/�, adelayed input gives a correspondingly delayed output, i.e.,

y.t � t0/ D H�x.t � t0/

�Impulse Response and Superposition Integral

� The impulse response of an LTI system is denoted

h.t/�D H

�ı.t/

�assuming the system is initially at rest

� Suppose we can write x.t/ as

x.t/ D

NXnD1

˛nı.t � tn/


2.7. LINEAR TIME INVARIANT (LTI) SYSTEMS

� For an LTI system with impulse response h. /

y.t/ D

NXnD1

˛nh.t � tn/

� To develop the superposition integral we write

x.t/ D

Z1

�1

x.�/ı.t � �/ d�

' limN!1

NXnD�N

x.n�t/ı.t � n�t/�t; for �t � 1

t

x(t)

0 ∆t−∆t 2∆t 3∆t 4∆t 5∆t 6∆t

Rectangle area is approximation

. . . . . .

Impulse sequence approximation to x.t/

� If we applyH to both sides and let�t ! 0 such that n�t ! �

we have

y.t/ ' limN!1

NXnD�N

x.n�t/h.t � n�t/�t

D

Z1

�1

x.�/h.t � �/ d� D x.t/ � h.t/

orD

Z1

�1

x.t � �/h.�/ d� D h.t/ � x.t/



2.7.1 Stability

� In signals and systems the concept of bounded-input bounded-output (BIBO) stability is introduced

� Satisfying this definition requires that every bounded-input (jx.t/j <1) produces a bounded output (jy.t/j <1)

� For LTI systems a fundamental theorem states that a system isBIBO stable if and only ifZ

1

�1

jh.t/j dt <1

� Further implications of this will be discussed later

2.7.2 Transfer Function

� The frequency domain result corresponding to the convolutionexpression y.t/ D x.t/ � h.t/ is

Y.f / D X.f /H.f /

where H.f / is known as the transfer function or frequencyresponse of the system having impulse response h.t/

� It immediately follows that

h.t/F ! H.f /

and

y.t/ D F�1˚X.f /H.f /

D

Z1

�1

X.f /H.f /ej 2�f t df



2.7.3 Causality

� A system is causal if the present output relies only on past andpresent inputs, that is the output does not anticipate the input

� The fact that for LTI systems y.t/ D x.t/ � h.t/ implies thatfor a causal system we must have

h.t/ D 0; t < 0

– Having h.t/ nonzero for t < 0 would incorporate futurevalues of the input to form the present value of the output

� Systems that are causal have limitations on their frequency re-sponse, in particular the Paley–Wiener theorem states that forR1

�1jh.t/j2 dt <1, H.f / for a causal system must satisfyZ

1

�1

j ln jH.f /jj1C f 2

df <1

� In simple terms this means:

1. We cannot have jH.f /j D 0 over a finite band of fre-quencies (isolated points ok)

2. The roll-off rate of jH.f /j cannot be too great, e.g., e�k1jf j

and e�k2jf j2

are not allowed, but polynomial forms suchasp1=.1C .f =fc/2N , N an integer, are acceptable

3. Practical filters such as Butterworth, Chebyshev, and el-liptical filters can come close to ideal requirements



2.7.4 Properties of H.f /

� For h.t/ real it follows that

jH.�f /j D jH.f /j and †H.�f / D �†H.f /

why?

� Input/output relationships for spectral densities are

Gy.f / D jY.f /j2D jX.f /H.f /j2 D jH.f /j2Gx.f /

Sy.f / D jH.f /j2Sx.f / proof in text chap. 6

Example 2.23: RC Lowpass Filter

C

R

x(t)

X(f ) Y(f )

y(t)

h(t), H(f)

ic(t)

vc(t)

RC lowpass filter schematic

� To find H.f / we may solve the circuit using AC steady-stateanalysis

Y.j!/

X.j!/D

1j!c

RC 1j!c

D1

1C j!RC

so

H.f / DY.f /

X.f /D

1

1C jf=f3; where f3 D 1=.2�RC/



� From the circuit differential equation

x.t/ D ic.t/RC y.t/

but

ic.t/ D cdvc.t/

dtD c

y.t/

dt

thus

RCdy.t/

dtC y.t/ D x.t/

� FT both sides using dx=dtF ! j 2�fX.f /

j 2�fRCY.f /C Y.f / D X.f /

so again

H.f / DY.f /

X.f /D

1

1C jf=f3

D1p

1C .f =f3/2e�j tan�1.f =f3/

� The Laplace transform could also be used here, and perhaps ispreferred, we just need to substitute s ! j! ! j 2�f



f

ff3

-f3

π/2

-π/2

1

RC lowpass frequency response

� Find the system response to

x.t/ D A…

�.t � T=2/

T

�� Finding Y.f / is easy since

Y.f / D X.f /H.f / D AT sinc.f T /�

1

1C jf=fs

�e�j�f t

� To find y.t/ we can IFT the above, use Laplace transforms, orconvolve directly

� From the FT tables we known that

h.t/ D1

RCe�t=.RC/ u.t/



� In Example 2.18 we showed that

Au.t/ � e�˛tu.t/ DA

˛

�1 � e�˛t

�u.t/

� Note that

A…

�t � T=2

T

�D AŒu.t/ � u.t � T /�

and here ˛ D 1=.RC/, so

y.t/ DA

RCRC

�1 � e�t=.RC/

�u.t/

�A

RCRC

�1 � e�.t�T /=.RC/

�u.t � T /

0.5 1 1.5 2 2.5 3

0.2

0.4

0.6

0.8

1

-3 -2 -1 1 2 3

0.2

0.4

0.6

0.8

1

-3 -2 -1 1 2 3

0.2

0.4

0.6

0.8

1

-3 -2 -1 1 2 3

0.2

0.4

0.6

0.8

1

t/T fT

fT fT

y(t) |X(f)|, |H(f)|, |Y(f)|

|X(f)|, |H(f)|, |Y(f)| |X(f)|, |H(f)|, |Y(f)|

RC = 2T

RC = T/2RC = T/10

T/10

RC =

T/5

T/2

T

2T

Pulse time response and frequency spectra with A D 1



2.7.5 Response to Periodic Inputs

� When the input is periodic we can write

x.t/ D

1XnD�1

Xnej 2�.nf0/t

which implies that

X.f / D

1XnD�1

Xnı.f � nf0/

� It then follows that

Y.f / D

1XnD�1

XnH.nf0/ı.f � nf0/

and

y.t/ D

1XnD�1

XnH.nf0/ej 2�.nf0/t

D

1XnD�1

jXnjjH.nf0jej Œ2�.nf0/tC†XnC†H.nf0/�

� This is a steady-state response calculation, since the analysisassumes that the periodic signal was applied to the system att D �1

2.7.6 Distortionless Transmission

� In the time domain a distortionless system is such that for anyinput x.t/,

y.t/ D H0x.t � t0/

where H0 and t0 are constants



� In the frequency domain the implies a frequency response ofthe form

H.f / D H0e�j 2�f t0;

that is the amplitude response is constant and the phase shift islinear with frequency

� Distortion types:

1. Amplitude response is not constant over a frequency band(interval) of interest$ amplitude distortion

2. Phase response is not linear over a frequency band of in-terest$ phase distortion

3. The system is non-linear, e.g., y.t/ D k0 C k1x.t/ C

k2x2.t/$ nonlinear distortion

2.7.7 Group and Phase Delay

� The phase distortion of a linear system can be characterizedusing group delay, Tg.f /,

Tg.f / D �1

2�

d�.f /

df

where �.f / is the phase response of an LTI system

� Note that for a distortionless system �.f / D �2�f t0, so

Tg.f / D �1

2�

d

df� 2�f t0 D t0 s;

clearly a constant group delay



� Tg.f / is the delay that two or more frequency componentsundergo in passing through an LTI system

– If say Tg.f1/ ¤ Tg.f2/ and both of these frequencies arein a band of interest, then we know that delay distortionexists

– Having two different frequency components arrive at thesystem output at different times produces signal disper-sion

� An LTI system passing a single frequency component, x.t/ DA cos.2�f1t /, always appears distortionless since at a singlefrequency the output is just

y.t/ D AjH.f1/j cos�2�f1t C �.f1/

�D AjH1.f /j cos

�2�f1

�t ��.f1/

2�f1

��which is equivalent to a delay known as the phase delay

Tp.f / D ��.f /

2�f

� The system output now is

y.t/ D AjH.f1/j cos�2�f1.t � Tp.f1//�

� Note that for a distortionless system

Tp.f / D �1

2�f.�2�f t0/ D t0



Example 2.24: Terminated Lossless Transmission Line

x(t) y(t)

Rs = R

0

RL = R

0R

0, v

p

L

y.t/ D1

2x�t �

L

vp

�Lossless transmission line

� We conclude that H0 D 1=2 and t0 D L=vp

� Note that a real transmission line does have losses that intro-duces dispersion on a wideband signal



Example 2.25: Fictitious System

-20 -10 10 20

0.5

1

1.5

2

-20 -10 10 20

-1.5

-1

-0.5

0.5

1

1.5

-20 -10 10 20

0.0025

0.005

0.0075

0.01

0.0125

0.015

-20 -10 10 20

0.011

0.012

0.013

0.014

0.015

0.016

f (Hz)

f (Hz)

f (Hz)f (Hz)

H(f)|H(f)|

Tg(f) T

p(f)

No distortion on |f | < 10 Hz band

Ampl. Radians

Time Time

Amplitude, phase, group delay, phase delay

� The system in this example is artificial, but the definitions canbe observed just the same

� For signals with spectral content limited to jf j < 10 Hz thereis no distortion, amplitude or phase/group delay

� For 10 < jf j < 15 amplitude distortion is present

� For jf j > 15 both amplitude and phase distortion are present

� What about the interval 10 < jf j < 15?



2.7.8 Nonlinear Distortion

� In the time domain a nonlinear system may be written as

y.t/ D

1XnD0

anxn.t/

� Specifically consider

y.t/ D a1x.t/C a2x2.t/

� Let

x.t/ D A1 cos.!1t /C A2 cos.!2t /

� Expanding the output we have

y.t/ D a1�A1 cos.!1t /C A2 cos.!2t /

�C a1

�A1 cos.!1t /C A2 cos.!2t /

�2D a1

�A1 cos.!1t /C A2 cos.!2t /

�C

na22

�A21 C A

22

�Ca2

2

�A21 cos.2!1t /C A22 cos.2!2t /

�oC a2A1A2

˚cosŒ.!1 C !2/t �C cosŒ.!1 � !2/t �

– The third line is the desired output

– The fourth line is termed harmonic distortion

– The fifth line is termed intermodulation distortion



f

f f

f

A1

A1

A2

a1A

1

a1A

1

a1A

1A

2

a1A

2a

1a

2A

1

a2A

1

2

2

a2A

1

2

2

a2A

2

2

2a

2(A

1 + A

2)

2 2

2

a2A

1

2

2

2f1

f2

f1

f1

2f1

2f2

f1

f1 f

2-f

1

f1+f

2

f2

0

0

0

0

Non-Linear

Non-Linear

Input

Input

Output

Output

One and two tones in y.t/ D a1x.t/C a2x2.t/ device

� In general if y.t/ D a1x.t/C a2x2.t/ the multiplication theo-rem implies that

Y.f / D a1X.f /C a2X.f / �X.f /

� In particular if X.f / D A…�f=.2W /

�

Y.f / D a1A…

�f

2W

�C a22WA

2ƒ

�f

2W



Y( f ) =

=

+f f

f

a1A 2Wa

2A2

Wa2A2

-W

-W

W

W

-2W

-2W

2W

2W

a1A + Wa

2A2

a1A + 2Wa

2A2

Continuous spectrum in y.t/ D a1x.t/C a2x2.t/ device

2.7.9 Ideal Filters

1. Lowpass of bandwidth B

HLP.f / D H0…

�f

2B

�e�j 2�f t0

B B-B -B

H0

slope =-2πt

0

|HLP

(f)| HLP

(f)

f f

2. Highpass with cutoff B

HHP.f / D H0

�1 �….f=.2B//

�e�j 2�f t0

B B-B -B

H0

slope =-2πt

0

|HHP

(f)|

f f

HHP

(f)



3. Bandpass of bandwidth B

HBP.f / D�Hl.f � f0/CHl.f C f0/

�e�j 2�f t0

where Hl.f / D H0….f=B/

B BH

0

slope = -2πt0

|HBP

(f)| HBP

(f)

f f-f

0f0

-f0

f0

� The impulse response of the lowpass filter is

hLP.t/ D F�1˚H0….f=.2B//e

�j 2�f t0

D 2BH0sincŒ2B.t � t0/�

� Ideal filters are not realizable, but simplify calculations andgive useful performance upper bound results

– Note that hLP.t/ ¤ 0 for t < 0, thus the filter is noncausaland unrealizable

� From the modulation theorem it also follows that

hBP.t/ D 2hl.t � t0/ cosŒ2�f0.t � t0/�D 2BH0sincŒB.t � t0/� cosŒ2�f0.t � t0/�



t t

hLP

(t) hBP

(t)

t0

t0

2BH0 2BH

0

t0 - 1

2B t0 - 1

2Bt0 + 1

2B t0 + 1

2B

Ideal lowpass and bandpass impulse responses

2.7.10 Realizable Filters

� We can approximate ideal filters with realizable filters such asButterworth, Chebyshev, and Bessel, to name a few

� We will only consider the lowpass case since via frequencytransformations we can obtain the others

Butterworth

� A Butterworth filter has a maximally flat (flat in the sense ofderivatives of the amplitude response at dc being zero) pass-band

� In the s-domain (s D �Cj!) the transfer function of a lowpassdesign is

HBU.s/ D!nc

.s � s1/.s � s2/ � � � .s � sn/

where

sk D !c exp��

�1

2C2k � 1

2n

��; k D 1; 2; : : : ; n



� Note that the poles are located on a semi-circle of radius !c D2�fc, where fc is the 3dB cuttoff frequency of the filter

� The amplitude response of a Butterworth filter is simply

jHBU.f /j1p

1C .f =fc/2n

Butterworth n D 4 lowpass filter

Chebyshev

� A Chebyshev type I filter (ripple in the passband), is is de-signed to maintain the maximum allowable attenuation in thepassband yet have maximum stopband attenuation

� The amplitude response is given by

jHC.f /j D1p

1C �2C 2n .f /

where

Cn.f / D

(cos.n cos�1.f =fc//; 0 � jf j � fc

cosh.n cosh�1.f =fc//; jf j > fc



� The poles are located on an ellipse as shown below

Chebyshev n D 4 lowpass filter

Bessel

� A Bessel filter is designed to maintain linear phase in the pass-band at the expense of the amplitude response

HBE.f / DKn

Bn.f /

where Bn.f / is a Bessel polynomial of order n (see text) andKn is chosen so that the filter gain is unity at f D 0



Amplitude Rolloff and Group Delay Comparision

� Compare Butterworth, 0.1 dB ripple Chebyshev, and Bessel

n D 3 Amplitude response

n D 3 Group delay



Filter Construction TechniquesConstructionType

Description of El-ements or Filter

Center Fre-quency Range

Unloaded Q

(typicalFilter Appli-cation

LC (passive) lumped elements DC–300 MHzor higher in in-tegrated form

100 Audio, video,IF and RF

Active R, C , op-amps DC–500 kHzor higher usingWB op-amps

200 Audio and lowRF

Crystal quartz crystal 1kHz – 100MHz

100,000 IF

Ceramic ceramic disks withelectrodes

10kHz – 10.7MHz

1,000 IF

Surface acousticwaves (SAW)

interdigitated fin-gers on a Piezo-electric substrate

10-800 MHz, variable IF and RF

Transmission line quarterwave stubs,open and short ckt

UHF and mi-crowave

1,000 RF

Cavity machined andplated metal

Microwave 10,000 RF

2.7.11 Pulse Resolution, Risetime, and Band-width

Problem: Given a non-bandlimited signal, what is a reasonable esti-mate of the signals transmission bandwidth?We would like to obtain a relationship to the signals time duration

� Step 1: We first consider a time domain relationship by seekinga constant T such that

T x.0/ D

Z1

�1

jx.t/j dt

t

x(0)

|x(t)|

T/2-T/2 0

Make areas equal via T



� Note thatZ1

�1

x.t/ dt D

Z1

�1


ˇ̌̌̌fD0

D X.0/

and Z1

�1

jx.t/j dt �

Z1

�1

x.t/ dt

which impliesT x.0/ � X.0/

� Step 2: Find a constant W such that

2WX.0/ D

Z1

�1

jX.f /j df

f

X(0)

|X(f)|

W-W 0

Make areas equal via W

� Note thatZ1

�1

X.f / df D

Z1

�1

X.f /ej 2�f t df

ˇ̌̌̌tD0

D x.0/

and Z1

�1

jX.f /j df �

Z1

�1

X.f / df

which implies that

2WX.0/ � x.0/



� Combining the results of Step 1 and Step 2, we have

2WX.0/ � x.0/ �1

TX.0/

or

2W �1

Tor W �

1

2T

Example 2.26: Rectangle Pulse

� Consider the pulse x.t/ D ….t=T /

� We know that X.f / D T sinc.f T /

-1 -0.5 0.5 1

0.2

0.4

0.6

0.8

1

-3 -2 -1 1 2 3

0.2

0.4

0.6

0.8

1

fTt /T

x(t) |X(f)|/T

f1/(2T)-1/(2T)

Lower bound for W

Pulse width versus Bandwidth, is W � 1=.2T ?

� We see that for the case of the sinc. / function the bandwidth,W , is clearly greater than the simple bound predicts



Risetime

� There is also a relationship between the risetime of a pulse-likesignal and bandwidth

� Definition: The risetime, TR, is the time required for the lead-ing edge of a pulse to go from 10% to 90% of its final value

� Given the impulse response h.t/ for an LTI system, the stepresponse is just

ys.t/ D

Z1

�1

h.�/u.t � �/ d�

D

Z t

�1

h.�/ d�if causalD

Z t

0

h.�/ d�

Example 2.27: Risetime of RC Lowpass

� The RC lowpass filter has impulse response

h.t/ D1

RCe�t=.RC/u.t/

� The step response is

ys.t/ D�1 � e�t=.RC/

�u.t/

� The risetime can be obtained by setting ys.t1/ D 0:1 and ys.t2/ D0:9

0:1 D�1 � e�t1=.RC/

�) ln.0:9/ D

�t1

RC

0:9 D�1 � e�t2=.RC/

�) ln.0:1/ D

�t2

RC



� The difference t2 � t1 is the risetime

TR D t2 � t1 D RC ln.0:9=0:1/ ' 2:2RC D0:35

f3

where f3 is the RC lowpass 3dB frequency

Example 2.28: Risetime of Ideal Lowpass

� The risetime of an ideal lowpass filter is of interest since it isused in modeling and also to see what an ideal filter does to astep input

� The impulse response is

h.t/ D F�1�…

�f

2B

��D 2BsincŒ2Bt�

� The step response then is

ys D

Z t

�1

2BsincŒ2B�� d�

D1

�

Z 2�Bt

�1

sinuu

du

D1

2C1

�SiŒ2�Bt�

where Si( ) is a special function known as the sine integral

� We can numerically find the risetime to be

TR '0:44

B



1 2 3 4 5

0.2

0.4

0.6

0.8

1

-2 -1 1 2 3

0.2

0.4

0.6

0.8

1

t RC t/B

Step Response of RC Lowpass Step Response of Ideal Lowpass

2.2 0.44

RC and ideal lowpass risetime comparison


2.8. SAMPLING THEORY

2.8 Sampling Theory

Integrate with Chapter 3 material.

2.9 The Hilbert Transform

Integrate with Chapter 3 material.

2.10 The Discrete Fourier Transform andFFT

?



.


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