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Discrete Systems & Z-Transforms
© 2014 School of Information Technology and Electrical Engineering at The University of Queensland
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http://elec3004.org
Lecture Schedule: Week Date Lecture Title
1 2-Mar Introduction
3-Mar Systems Overview
2 9-Mar Signals as Vectors & Systems as Maps
10-Mar [Signals]
3 16-Mar Sampling & Data Acquisition & Antialiasing Filters
17-Mar [Sampling]
4 23-Mar System Analysis & Convolution
24-Mar [Convolution & FT]
5 30-Mar Discrete Systems & Z-Transforms 31-Mar [Z-Transforms]
6 13-Apr Frequency Response & Filter Analysis
14-Apr [Filters]
7 20-Apr Digital Filters
21-Apr [Digital Filters]
8 27-Apr Introduction to Digital Control
28-Apr [Feedback]
9 4-May Digital Control Design
5-May [Digitial Control]
10 11-May Stability of Digital Systems
12-May [Stability]
11 18-May State-Space
19-May Controllability & Observability
12 25-May PID Control & System Identification
26-May Digitial Control System Hardware
13 31-May Applications in Industry & Information Theory & Communications
2-Jun Summary and Course Review
30 March 2015 - ELEC 3004: Systems 2
2
Convolution ℱ: Fourier Series
(Periodic functions)
ℒ ℱ: (𝜉 = 𝜎 + 𝑖𝜏)
(ℝ ℂ) ℂ: Poles & Zeros DFFT Z-Transform
Lecture Overview
ODE
ℒ: Laplace (s)
Transfer functions
Cascade of LCC ODE
Convolution
Z-Transform
• Course So Far:
• Lecture(s):
30 March 2015 - ELEC 3004: Systems 3
For c(τ)= :
1. Keep the function f (τ) fixed
2. Flip (invert) the function g(τ) about the vertical axis (τ=0)
= this is g(-τ)
3. Shift this frame (g(-τ)) along τ (horizontal axis) by t0.
= this is g(t0 -τ)
For c(t0):
4. c(t0) = the area under the product of f (τ) and g(t0 -τ)
5. Repeat this procedure, shifting the frame by different values
(positive and negative) to obtain c(t) for all values of t.
Graphical Understanding of Convolution
30 March 2015 - ELEC 3004: Systems 4
3
Graphical Understanding of Convolution (Ex)
30 March 2015 - ELEC 3004: Systems 5
Complex Numbers and Phasors
Y
X
Re j
R
Rcos( )
Rsin( )
Re ( cos , sin )
cos sin
(cos sin )
j R R
R jR
R j
Positive Frequency
component
30 March 2015 - ELEC 3004: Systems 6
4
Complex Numbers and Phasors
Y
X
Re j
R
R cos( )
Rsin( )
Re ( cos( ), sin( ))
cos( ) sin( )
(cos sin )
j R R
R jR
R j
Negative frequency
component
30 March 2015 - ELEC 3004: Systems 7
Positive and Negative Frequencies • Frequency is the derivative of phase
more nuanced than : 1
𝜏= 𝑟𝑒𝑝𝑒𝑡𝑖𝑡𝑖𝑜𝑛 𝑟𝑎𝑡𝑒
• Hence both positive and negative frequencies are possible.
• Compare – velocity vs speed
– frequency vs repetition rate
30 March 2015 - ELEC 3004: Systems 8
5
• Q: What is negative frequency?
• A: A mathematical convenience
• Trigonometrical FS – periodic signal is made up from
– sum 0 to of sine and cosines ‘harmonics’
• Complex Fourier Series & the Fourier Transform – use exp(jωt) instead of cos(ωt) and sin(ωt)
– signal is sum from 0 to of exp(jωt)
– same as sum - to of exp(-jωt)
– which is more compact (i.e., less chalk!)
Negative Frequency
30 March 2015 - ELEC 3004: Systems 9
Rotating wheel and peg
Top
View
Front
View
Need both top and front
view to determine rotation
Another way to see Aliasing Too!
30 March 2015 - ELEC 3004: Systems 10
6
Frequency Response
Fourier Series Fourier Transforms
30 March 2015 - ELEC 3004: Systems 11
Typical Linear Processors • Convolution h(n,k)=h(n-k)
• Cross Correlation h(n,k)=h(n+k)
• Auto Correlation h(n,k)=x(k-n)
• Cosine Transform h(n,k)=
• Sine Transform h(n,k)=
• Fourier Transform h(n,k)=
cos2
Nnk
sin2
Nnk
exp jN
nk2
30 March 2015 - ELEC 3004: Systems 12
7
• Signal measured (or known) as a function of an independent
variable – e.g., time: y = f(t)
• However, this independent variable may not be the most
appropriate/informative – e.g., frequency: Y = f(w)
• Therefore, need to transform from one domain to the other – e.g., time frequency
– As used by the human ear (and eye)
Transform Analysis
Signal processing uses Fourier, Laplace, & z transforms etc
30 March 2015 - ELEC 3004: Systems 13
Sinusoids and Linear Systems
If
or
x t A t( ) cos( ) 0 0
x n A nt( ) cos( ) 0 0
then in steady state
h(t) or h(n)
x(t) or x(n) y(t) or y(n)
y t AC t( ) ( )cos( ( )) 0 0 0 0
y n AC T nt T( ) ( )cos( ( )) 0 0 0 0
30 March 2015 - ELEC 3004: Systems 14
8
• The pair of numbers C(ω0) and q(ω0) are the complex gain of
the system at the frequency ω0 .
• They are respectively, the magnitude response and the phase
response at the frequency ω0 .
Sinusoids and Linear Systems
y t AC t( ) ( )cos( ( )) 0 0 0 0
y n AC T nt T( ) ( )cos( ( )) 0 0 0 0
30 March 2015 - ELEC 3004: Systems 15
• Why probe system with sinusoids?
• Sinusoids are eigenfunctions of linear systems???
• What the hell does that mean?
• Sinusoid in implies sinusoid out
• Only need to know phase and magnitude (two parameters) to
fully describe output rather than whole waveform – sine + sine = sine
– derivative of sine = sine (phase shifted - cos)
– integral of sine = sine (-cos)
• Sinusoids maintain orthogonality after sampling (not true of
most orthogonal sets)
Why Use Sinusoids?
30 March 2015 - ELEC 3004: Systems 16
9
Frequency Response
Fourier Series Fourier Transforms
30 March 2015 - ELEC 3004: Systems 17
Fourier Series
• Deal with continuous-time periodic signals.
• Discrete frequency spectra.
A Periodic Signal
T 2T 3T
t
f(t)
Source: URI ELE436
30 March 2015 - ELEC 3004: Systems 18
10
Two Forms for Fourier Series
T
ntb
T
nta
atf
n
n
n
n
2sin
2cos
2)(
11
0Sinusoidal
Form
Complex
Form:
n
tjn
nectf 0)( dtetfT
cT
T
tjn
n
2/
2/
0)(1
dttfT
aT
T2/
2/0 )(
2tdtntf
Ta
T
Tn 0
2/
2/cos)(
2
tdtntfT
bT
Tn 0
2/
2/sin)(
2
Source: URI ELE436
30 March 2015 - ELEC 3004: Systems 19
• Any finite power, periodic, signal x(t) – period T
• can be represented as () summation of – sine and cosine waves
• Called: Trigonometrical Fourier Series
Fourier Series
00 0
1
( ) cos( ) sin( )2
n n
n
Ax t A n t B n t
• Fundamental frequency ω0=2/T rad/s or 1/T Hz
• DC (average) value A0 /2
30 March 2015 - ELEC 3004: Systems 20
11
0 1 2 3 4 5 6 -2
-1
0
1
2
time (t)
y =
f(t
)
0 1 2 3 4 5 6 7 8 0
0.5
1
1.5
frequency (f)
Am
plit
ude
Frequency representation (spectrum) shows signal contains:
• 2Hz and 5Hz components (sinewaves) of equal amplitude
30 March 2015 - ELEC 3004: Systems 21
• An & Bn calculated from the signal, x(t) – called: Fourier coefficients
Fourier Series Coefficients
,3,2,1)sin()(2
,2,1,0)cos()(2
2/
2/
0
2/
2/
0
ndttnwtxT
B
ndttnwtxT
A
T
T
n
T
T
n
Note: Limits of integration can vary,
provided they cover one period
30 March 2015 - ELEC 3004: Systems 22
12
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 -1.5
-1
-0.5
0
0.5
1
1.5
time (s)
signal (black), approximation (red) and harmonic (green)
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 -1.5
-1
-0.5
0
0.5
1
1.5
time (s)
harm
onic
30 March 2015 - ELEC 3004: Systems 23
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 -1.5
-1
-0.5
0
0.5
1
1.5
time (s)
signal (black), approximation (red) and harmonic (green)
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 -1.5
-1
-0.5
0
0.5
1
1.5
time (s)
harm
onic
30 March 2015 - ELEC 3004: Systems 24
13
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 -1.5
-1
-0.5
0
0.5
1
1.5
time (s)
signal (black), approximation (red) and harmonic (green)
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 -1.5
-1
-0.5
0
0.5
1
1.5
time (s)
harm
onic
30 March 2015 - ELEC 3004: Systems 25
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 -1.5
-1
-0.5
0
0.5
1
1.5
time (s)
signal (black), approximation (red) and harmonic (green)
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 -1.5
-1
-0.5
0
0.5
1
1.5
time (s)
harm
onic
30 March 2015 - ELEC 3004: Systems 26
14
Example: Square wave
))cos(1(2
cos11coscoscos
)sin()sin()sin()(
0sinsin
)cos()cos()cos()(
).2(
;21,1
;10,1
)(
2
1
1
0
2
1
1
0
2
0
2
1
1
0
2
1
1
0
2
0
nn
B
n
n
nnn
n
n
tn
n
tnB
dttndttndttntxB
n
tn
n
tnA
dttndttndttntxA
tx
t
t
tx
n
n
n
n
n
periodic! i.e., x(t + 2) = x(t)
No cos terms as sin(n) = 0 n
x(t) has odd symmetry
Sin terms only
cos(2n) = 1 n
30 March 2015 - ELEC 3004: Systems 28
Example: Square wave
etc
harmonic)(fifth )5sin(5
4
harmonic)(fourth 0
harmonic) (third)3sin(3
4
harmonic) (second0
al)(fundament)sin(4
x(t)
gives, terms theExpanding
)sin())cos(1(2
)(
is, seriesFourier ricTrigonomet Therefore,
1
t
t
t
tnnn
txn
• Only odd harmonics;
• In proportion
1,1/3,1/5,1/7,…
• Higher harmonics
contribute less;
• Therefore, converges
30 March 2015 - ELEC 3004: Systems 29
15
How to Deal with Aperiodic Signal?
A Periodic Signal
T
t
f(t)
If T, what happens? Source: URI ELE436
30 March 2015 - ELEC 3004: Systems 30
tjn
n
nT ectf 0)(
Fourier Integral
n
tjnT
T
jn
T edefT
002/
2/)(
1
dtetfT
cT
T
tjn
Tn
2/
2/
0)(1
T
20
2
1 0
T
n
tjnT
T
jn
T edef 00
0
2/
2/)(
2
1
Let T
20
0 dT
n
tjnT
T
jn
T edef 002/
2/)(
2
1
dedef tjj
T )(2
1
Source: URI ELE436
30 March 2015 - ELEC 3004: Systems 31
16
dedeftf tjj)(2
1)(
Fourier Integral
F(j)
dtetfjF tj
)()(
dejFtf tj)(2
1)( Synthesis
Analysis
Source: URI ELE436
30 March 2015 - ELEC 3004: Systems 32
Fourier Series vs. Fourier Integral
n
tjn
nectf 0)(Fourier
Series:
Fourier
Integral:
dtetfT
cT
T
tjn
Tn
2/
2/
0)(1
dtetfjF tj
)()(
dejFtf tj)(2
1)(
Period Function
Discrete Spectra
Non-Period
Function
Continuous Spectra
Source: URI ELE436
30 March 2015 - ELEC 3004: Systems 33
17
Complex Fourier Series (CFS)
• Also called Exponential
Fourier series
– As it uses Euler’s relation
• FS as a Complex phasor
summation
j
tjnwtjnwtnw
tjnwtjnwtnw
twjAtwAtjwA
2
)exp()exp()sin(
2
)exp()exp()cos(
implies,which
)sin()cos()exp(
000
000
000
n
n tjnwXtx )exp()( 0Where Xn are the
CFS coefficients
30 March 2015 - ELEC 3004: Systems 34
Complex Fourier Coefficients
• Again, Xn calculated from
x(t)
• Only one set of coefficients,
Xn
– but, generally they are
complex
2/
2/
0 )exp()(1
T
T
n dttjnwtxT
X
Remember: fundamental ω0 = 2/T !
30 March 2015 - ELEC 3004: Systems 35
18
Relationships
• There is a simple
relationship between
– trigonometrical and
– complex Fourier
coefficients,
.0,2
;0,2
2
00
njBA
njBA
X
AX
nn
nn
n
Therefore, can calculate simplest form and convert
Constrained to be
symmetrical, i.e.,
complex conjugate
*
nn XX
30 March 2015 - ELEC 3004: Systems 36
Example: Complex FS
• Consider the pulse train
signal
• Has complex Fourier series:
).(
;2
,0
;2
0,
)(
Ttx
Tt
tA
tx
T
2exp
2exp
exp1
exp)(1
00
0
2
2
0
2
2
0
jnjn
Tjn
A
dttjnAT
dttjntxT
X
T
T
n
A
Note: n is the
ind. variable
Note: ×
by / …
30 March 2015 - ELEC 3004: Systems 37
19
Example: Complex FS
• Which using Euler’s identity
reduces to:
Tw
nwT
A
nw
nw
T
AX n
2
)2(sa2
)2sin(
0
0
0
0
sin2sincossincos
sincossincos
expexp
jjj
jj
jj
Note: letting 2
0
n
Note:
cos(-) = cos(): even
sin(-) = -sin(): odd
30 March 2015 - ELEC 3004: Systems 38
Frequency Response
Fourier Series Fourier Transforms
30 March 2015 - ELEC 3004: Systems 54
20
• A Fourier Transform is an integral transform that re-expresses
a function in terms of different sine waves of varying
amplitudes, wavelengths, and phases.
• When you let these three waves interfere with each other you
get your original wave function!
Fourier Transform
1-D Example:
Source: Tufts Uni Sykes Group
30 March 2015 - ELEC 3004: Systems 55
Fourier Series • What we have produced is a processor to calculate one
coefficient of the complex Fourier Series
• Fourier Series Coefficients = Heterodyne and average over
observation interval T
CT
h t e dtk
jT
ktT
1
2
0
( )
30 March 2015 - ELEC 3004: Systems 56
21
• If we change the limits of integration to the entire real line,
remove the division by T, and make the frequency variable
continuous, we get the Fourier Transform
ELEC 3004: Systems 30 March 2015 - 57
Fourier Transform
C h t e dtj t( ) ( )
Fourier Transform (is not the Fourier Series per se)
Fourier
Series
Discrete
Fourier
Transform
Continuous
Fourier
Transform
Fourier
Transform
Continuous
Time
Discrete
Time
Per
iodic
A
per
iodic
Source: URI ELE436
30 March 2015 - ELEC 3004: Systems 58
22
• Fourier series – Only applicable to periodic signals
• Real world signals are rarely periodic
• Develop Fourier transform by – Examining a periodic signal
– Extending the period to infinity
Fourier Transform
30 March 2015 - ELEC 3004: Systems 59
• Problem: as T , Xn 0 – i.e., Fourier coefficients vanish!
• Solution: re-define coefficients – Xn’ = T × Xn
• As T – (harmonic frequency) nω0 ω (continuous freq.)
– (discrete spectrum) Xn’ X(ω) (continuous spect.)
– ω0 (fundamental freq.) reduces dω (differential) • Summation becomes integration
Fourier Transform
30 March 2015 - ELEC 3004: Systems 60
23
Fourier Transform Pair
dtetfjF tj
)()(
dejFtf tj)(2
1)( Synthesis
Analysis
Fourier Transform:
Inverse Fourier Transform:
Source: URI ELE436
30 March 2015 - ELEC 3004: Systems 63
dtetfjF tj
)()(
Continuous Spectra
)()()( jjFjFjF IR
)(|)(| jejF FR(j)
FI(j)
()
Magnitude
Phase
Source: URI ELE436
30 March 2015 - ELEC 3004: Systems 64
24
-10 -8 -6 -4 -2 0 2 4 6 8 10 0
0.2
0.4
0.6
0.8
1
Pulse width = 1
x(t
)
time (t)
-20 -5 0 5 20 -0.5
0
0.5
1
X(w
)
Angular frequency (w)
rect(t)
sinc(w/2)
Time limited
Infinite bandwidth
30 March 2015 - ELEC 3004: Systems 66
-10 -8 -6 -4 -2 0 2 4 6 8 10 0
0.2
0.4
0.6
0.8
1
Pulse width = 2
x(t
)
time (t)
-10 -5 0 5 10 -0.5
0
0.5
1
1.5
2
X(w
)
Angular frequency (w)
rect(t/2)
2 sinc(w/)
Parseval’s Theorem
30 March 2015 - ELEC 3004: Systems 67
25
-10 -8 -6 -4 -2 0 2 4 6 8 10 0
0.2
0.4
0.6
0.8
1
Pulse width = 4
x(t
)
time (t)
-10 -5 0 5 10 -1
0
1
2
3
4
X(w
)
Angular frequency (w)
rect(t/4)
4 sinc(2w/)
30 March 2015 - ELEC 3004: Systems 68
-10 -8 -6 -4 -2 0 2 4 6 8 10 0
0.2
0.4
0.6
0.8
1
Pulse width = 8
x(t
)
time (t)
-10 -5 0 5 10 -2
0
2
4
6
8
X(w
)
Angular frequency (w)
rect(t/8)
8 sinc(4w/)
30 March 2015 - ELEC 3004: Systems 69
26
-40 -30 -20 -10 0 10 20 30 40 -0.25
0
0.5
1
Symmetry: F{sinc(t/2)} = 2 rect(-w)
x(t
)
time (t)
-5 -4 -3 -2 -1 0 1 2 3 4 5 0
2
X(w
)
Angular frequency (w)
2 rect(-w)
sinc(t/2)
Infinite time
Finite bandwidth
‘Ideal’ Lowpass filter 30 March 2015 - ELEC 3004: Systems 70
• Linearity – F {a x(t) + b y(t)} = a X(ω) + b Y(ω)
• Time and frequency scaling – F {x(at)} = 1/a X(ω/a)
– broader in time narrower in frequency • and vice versa
• Symmetry (duality) – 2x(-ω) = X(t) exp(-jωt)dt
• i.e., Fourier transform ‘pairs’
Properties of Fourier Transform
Time limited signal limited has infinite bandwidth;
Signal of finite bandwidth has infinite time support
30 March 2015 - ELEC 3004: Systems 71
27
Properties of Fourier Transform
• if…
• x(t) is real
• x(t) is real and even
• x(t) is real and odd
• Then…
• X(-ω) = X(ω)*
– {X(ω)} is even
– {X(ω)} is odd
– |X(ω)| is even
– X(ω) is odd
• X(ω) is real and even
• X(ω) is imaginary and odd
30 March 2015 - ELEC 3004: Systems 72
Fourier Transforms
-15 -10 -5 0 5 10 15
-1
-0.5
0
0.5
1
t
x(t
)
-ω0 0
ω0
0
1
2
3
ω
X(w
)
x(t) = cos(ω0t)
(real & even)
X(w) = [(ω-ω0)
+ (ω+ω0)]
(real and even)
1
0 0
1| | exp
2X x t F X x t j t
Note: cos(ω0t) has energy! But is dual of (w – ω0)
30 March 2015 - ELEC 3004: Systems 73
28
Fourier Transforms
x(t) = sin(ω0t)
(real and odd)
X(w) = j[(ω+ω0)
- (ω-ω0)]
(imaginary & odd)
-15 -10 -5 0 5 10 15
-1
-0.5
0
0.5
1
t
x(t
)
-ω0 0
ω0 3 4
-4
-2
0
2
4
{X
(w)}
Note: sin & cos have same Mag spectrum, Phase is only difference
30 March 2015 - ELEC 3004: Systems 74
• Time Shift – F {x(t - )} = exp(-j w)X(w)
• time shift phase shift
• Convolution and multiplication – F {x(t) y(t)} = X(ω) Y(ω)
• i.e., implement convolution in Fourier domain
– F {x(t) y(t)} = 1/2 (X(ω) Y(ω)) • i.e., Fourier interpretation of multiplication (e.g., frequency modulation)
Properties of Fourier Transform
30 March 2015 - ELEC 3004: Systems 75
29
-15 -10 -5 0 5 10 15 0
0.2
0.4
0.6
0.8
1
1.2
Frequency (rad/s)
|exp
(-j
)|
-15 -10 -5 0 5 10 15 -200
-100
0
100
200
Frequency (rad/s)
ph
ase
(d
egre
es)
Magnitude and Phase of exp(-j w)
modifies phase only
as
cos2 +
sin2 =1
30 March 2015 - ELEC 3004: Systems 76
0
0.5
1
x(t) = rect(t)
0
0.5
1
h(t) = rect(t)
-1 0 1 2 0
x(t)*h(t) = tri(t)
-0.5
0
0.5
1 X(w) = sinc(w/2)
-0.5
0
0.5
1 H(w) = sinc(w/2)
-20 -10 0 10 20 0
0.5
1 X(w)H(w) = sinc
2 (w/2)
F
F
F
ZOH
ZOH
FOH
30 March 2015 - ELEC 3004: Systems 77
30
• Differentiation in time
• Integration in time
More properties of the FT
n
n
dF x t j X
dt
dF x t j X
dt
Differentiation ×ω (Note: HPF & DC x zero)
1
0
t
F x t dt X Xj
Integration /ω + DC offset (LPF & opposite of differentiation)
30 March 2015 - ELEC 3004: Systems 78
More Fourier Transforms
ω
F(w) = (2/t)(ω - 2n/t)
t
f(t) =(t - nt) = (t)
… … … …
F
ω
F(w) = (2/t) (ω - n/t)
t
f(t) = (t - 2nt)
… … … …
F
Impulse train, ‘comb’ or ‘Shah’ function
30 March 2015 - ELEC 3004: Systems 79
31
More Fourier Transforms
t
f(t) = (t)
ω
F(ω) = 1
F
ω
F(ω) = 2(ω)
t
f(t) = 1
F
Limit of previous as t and t 0 respectively
Note: f(t) = 1 has energy! But is dual of (t)
Note: u(t) also has energy! But F{u(t)} = F{(t)} i.e., apply integration property
30 March 2015 - ELEC 3004: Systems 80
• If F{x(t)} = X(w) – F{x(2t)} =?
– F{x(t/4)} =?
• F{(t)} = ?
• F{1} = ?
“Pop Quiz”: Questions
30 March 2015 - ELEC 3004: Systems 81
32
… No Worries LectopiaLand !
• If F{x(t)} = X(ω) – F{x(2t)} = 1/2X(ω/2)
• narrower in t broader in freq
– F{x(t/4)} = 4X(4ω) • broader in t narrower in freq (but increased amplitude)
• F{(t)} = 1 – i.e. flat spectrum (all frequencies equally)
• F{1} = (ω) – i.e. impulse at DC only
Pop Quiz: Answers!
30 March 2015 - ELEC 3004: Systems 82
• Represents (usually finite energy) signals – as sum of cosine waves
• at all possible frequencies
• |X(ω)|dω/2 is amplitude of cosine wave – i.e., in frequency band ω to ω + dω
• X(ω) is phase shift of cosine wave
• Also represents finite power, periodic signals – Using (ω)
• Distribution with frequency of – both magnitude & phase
– called a Frequency spectrum (continuous)
Interpretation of Fourier Transform
30 March 2015 - ELEC 3004: Systems 83
33
Fourier Image Examples
Lena Bridge
30 March 2015 - ELEC 3004: Systems 84
Fourier Magnitude and Phase
20*log10(abs(fft(Lena))) angle(fft(Lena))
0 10 20 30 40 50 60 70 80 90 1000
0.5
1
Frequency
Magnitude
1/f
‘random’
range()
Bridge spectra look similar
30 March 2015 - ELEC 3004: Systems 85
34
Magnitude and Phase Only
ifft(abs(fft(Lena)) + angle(0)) ifft(abs(fft(Bridge)) + angle(fft(Lena)))
Lena magnitude only Lena phase + bridge magnitude
Note: titles are illustrative only and are not the actual Matlab commands used!
30 March 2015 - ELEC 3004: Systems 86
FFT Fourier Transform (not just yet – we’ll come back to this)
30 March 2015 - ELEC 3004: Systems 98
35
• For a discrete time sequence we define two classes of
Fourier Transforms:
• the DTFT (Discrete Time FT) for sequences having
infinite duration,
• the DFT (Discrete FT) for sequences having finite
duration.
Fourier Analysis of Discrete Time Signals
30 March 2015 - ELEC 3004: Systems 99
• Given a sequence x(n) having infinite duration, we define the
DTFT as follows:
The Discrete Time Fourier Transform (DTFT)
X DTFT x n x n e
x n IDTFT X X e d
j n
n
j n
( ) ( ) ( )
( ) ( ) ( )
1
2
x n( )
nN 1
X ( )
discrete time continuous
frequency
….. …..
30 March 2015 - ELEC 3004: Systems 100
36
Observations:
• The DTFT is periodic with period ;
• The frequency is the digital frequency and therefore it is limited to
the interval
)(X 2
Recall that the digital frequency is a normalized frequency relative to
the sampling frequency, defined as
sF
F 2
0
0
2/sF2/sFsF sF F
22
one period of )(X
30 March 2015 - ELEC 3004: Systems 101
Example
0 1N
1
][nx
n
2/sin
2/sin
1
1)(
2/)1(
1
0
Ne
e
eeX
Nj
j
NjN
n
nj
DTFT
since
30 March 2015 - ELEC 3004: Systems 102
37
Fast Fourier Transform Algorithms • Consider DTFT
• Basic idea is to split the sum into 2 subsequences of length N/2
and continue all the way down until you have N/2
subsequences of length 2
Log2(8)
N
30 March 2015 - ELEC 3004: Systems 103
Radix-2 FFT Algorithms - Two point FFT • We assume N=2^m
– This is called Radix-2 FFT Algorithms
• Let’s take a simple example where only two points are given n=0, n=1; N=2
Source: http://www.cmlab.csie.ntu.edu.tw/cml/dsp/training/coding/transform/fft.html
y0
y1
y0
Butterfly FFT
Advantage: Less
computationally
intensive: O(N/2*log(N))
30 March 2015 - ELEC 3004: Systems 104
38
• First break x[n] into even and odd
• Let n=2m for even and n=2m+1 for odd
• Even and odd parts are both DFT of a N/2 point sequence
• Break up the size N/2 subsequent in half by letting 2mm
• The first subsequence here is the term x[0], x[4], …
• The second subsequent is x[2], x[6], …
1
1)2sin()2cos(
)]12[(]2[
2/
2
2/
2/
2/2/
2/
2/
2/
2
12/
0
2/
12/
0
2/
N
N
jN
N
m
N
N
N
m
N
Nm
N
mk
N
mk
N
N
m
mk
N
k
N
N
m
mk
N
W
jeW
WWWW
WW
mxWWmxW
General FFT Algorithm
30 March 2015 - ELEC 3004: Systems 105
Example
1
1)2sin()2cos(
)]12[(]2[][
2/
2
2/
2/
2/2/
2/
2/
2/
2
12/
0
2/
12/
0
2/
N
N
jN
N
m
N
N
N
m
N
Nm
N
mk
N
mk
N
N
m
mk
N
k
N
N
m
mk
N
W
jeW
WWWW
WW
mxWWmxWkX
Let’s take a simple example where only two points are given n=0, n=1; N=2
]1[]0[]1[]0[)]1[(]0[]1[
]1[]0[)]1[(]0[]0[
1
1
0
0
1.0
1
1
1
0
0
1.0
1
0
0
0.0
1
0
1
0
0
0.0
1
xxxWxxWWxWkX
xxxWWxWkX
mm
mm
Same result
30 March 2015 - ELEC 3004: Systems 106
39
FFT Algorithms - Four point FFT
First find even and odd parts and then combine them:
The general form:
30 March 2015 - ELEC 3004: Systems 107
FFT Algorithms - 8 point FFT
http://www.engineeringproductivitytools.com/stuff/T0001/PT07.HTM
Applet:
http://www.falstad.com/fourier/directions.html
30 March 2015 - ELEC 3004: Systems 108
40
Poles and Zeros
30 March 2015 - ELEC 3004: Systems 109
Poles and Zeros
Source: Boyd, EE102,5-12
30 March 2015 - ELEC 3004: Systems 110
41
Poles and Zeros
Source: Boyd, EE102,5-13
30 March 2015 - ELEC 3004: Systems 111
Pole Zero Plot
Source: Boyd, EE102,5-14
30 March 2015 - ELEC 3004: Systems 112
42
Partial Fraction Expansion
Source: Boyd, EE102,5-15
30 March 2015 - ELEC 3004: Systems 113
Partial Fraction Expansion Example
Source: Boyd, EE102,5-16
30 March 2015 - ELEC 3004: Systems 114
43
How to Handle the Digitization?
(z-Transforms)
30 March 2015 - ELEC 3004: Systems 115
The z-transform
• The discrete equivalent is the z-Transform†:
𝒵 𝑓 𝑘 = 𝑓(𝑘)𝑧−𝑘∞
𝑘=0
= 𝐹 𝑧
and
𝒵 𝑓 𝑘 − 1 = 𝑧−1𝐹 𝑧
Convenient!
†This is not an approximation, but approximations are easier to derive
F(z) y(k) x(k)
30 March 2015 - ELEC 3004: Systems 116
44
The z-Transform
• It is defined by:
Or in the Laplace domain:
𝑧 = 𝑒𝑠𝑇
• Thus: or
• I.E., It’s a discrete version of the Laplace:
𝑓 𝑘𝑇 = 𝑒−𝑎𝑘𝑇 ⇒ 𝒵 𝑓 𝑘 =𝑧
𝑧 − 𝑒−𝑎𝑇
30 March 2015 - ELEC 3004: Systems 117
The z-transform • In practice, you’ll use look-up tables or computer tools (ie. Matlab)
to find the z-transform of your functions
𝑭(𝒔) F(kt) 𝑭(𝒛)
1
𝑠
1 𝑧
𝑧 − 1
1
𝑠2
𝑘𝑇 𝑇𝑧
𝑧 − 1 2
1
𝑠 + 𝑎
𝑒−𝑎𝑘𝑇 𝑧
𝑧 − 𝑒−𝑎𝑇
1
𝑠 + 𝑎 2
𝑘𝑇𝑒−𝑎𝑘𝑇 𝑧𝑇𝑒−𝑎𝑇
𝑧 − 𝑒−𝑎𝑇 2
1
𝑠2 + 𝑎2
sin (𝑎𝑘𝑇) 𝑧 sin𝑎𝑇
𝑧2− 2cos𝑎𝑇 𝑧 + 1
30 March 2015 - ELEC 3004: Systems 118
45
ℒ(ZOH)=??? : What is it?
• Lathi
• Franklin, Powell, Workman
• Franklin, Powell, Emani-Naeini
• Dorf & Bishop
• Oxford Discrete Systems:
(Mark Cannon)
• MIT 6.002 (Russ Tedrake)
• Matlab
Proof!
• Wikipedia
30 March 2015 - ELEC 3004: Systems 119
• Assume that the signal x(t) is zero for t<0, then the output
h(t) is related to x(t) as follows:
Zero-order-hold (ZOH)
x(t) x(kT) h(t) Zero-order
Hold Sampler
30 March 2015 - ELEC 3004: Systems 120
46
• Recall the Laplace Transforms (ℒ) of:
• Thus the ℒ of h(t) becomes:
Transfer function of Zero-order-hold (ZOH)
30 March 2015 - ELEC 3004: Systems 121
… Continuing the ℒ of h(t) …
Thus, giving the transfer function as:
Transfer function of Zero-order-hold (ZOH)
𝓩
30 March 2015 - ELEC 3004: Systems 122