Signal

EE-3423, SIGNALS & SYSTEMS I, ART GRIGORYAN 1

.

EE-3423 Signals and Systems I

Instructor: Dr. Artyom M. Grigoryan

EE Department, The University of Texas at San Antonio, Summer 2010


Abstract

Digital signal processing is an area of science and engineering that has developed rapidly over the past

thirty years. The rapid developed is a result of the significant advances in digital computer technology and

integrated-circuit fabrication.

Signal processing plays a central role in modern sciences and technology. Applications are founded

everywhere, including speech communication, acoustic, biomedical engineering, seismology, andmany others. The theory of signal processing is concerned in representation, transmission, and

manipulations of signals.Until to the 1960s the technology for signal processing has been carried in general by analog

methods. The evolution of digital computer and microprocessors has been pushed the developingdiscrete versions of signal processing methods. Digital signal processing becomes applicable in

many areas beginning from medical imaging to radar processing. That growth has created amassive amount of data, which is important to analyze, process, and transmit signals.

The digital signal and image processing have successful applications in geology, biology, meteo-rology, astronomy, radiolocation, television, medical diagnosis, and many others domains in scienceand engineering.

A. Modeling and Signals

We consider two topics of signals and systems as related to engineering:

1. Modeling of physical systems by mathematical equations.2. Modeling of physical signals by mathematical functions.

Model: It is advantageous to represent a device or an entire system by a circuit model. Forinstance, we can consider the 10m of copper wire is wound into the form of a multiturn coil and

that a voltage of variable frequency is applied. If the ratio of applied voltage V to resulting currentI is measured as a function of frequency. The model of this simple physical system is described by

the equation

Ldi(t)

dt+ Ri(t) +

1

C

∫τ<t

i(τ)dτ = v(t)

where R is the parameter of resistance, C is capacitance, and L is induction.Signals. Our word is full of signals, both natural and man-made. A signal can be defined as

a function that conveys information about the state or behavior of physical system. Signals are

represented mathematically as functions of one or more independent variables.For example, the functions f(t) = 2t + 1 and f(t) = 3t2 + 2t + 1 describe two signals, which vary

respectively linearly and quadratically with variable t.

Example of signals:

– The daily highs and lows in temperature. We can express the temperature at a designatedpoint in the room as a function θ(x, y, z, t) of four independent variables, where (x, y, z) are space

coordinates of the point and t is time.– The periodic electrical signals generated by the heart. Electrocardiogram (ECG) signal pro-

vides information about the activity of the patient’s heart. Electroencephalogram (EEG) signalprovides information about the activity of the brain. The variation is air pressure when we speak.

A speech can be represented mathematically as function of time.– The music we hear from compact disk (CD) player, which is due to changes in the air pressure

caused by the vibration of the speaker diaphragm. The information stored on CD is in a digital

form and, then, is converted to the analog form before we can hear the music.


– Seismic signal is shown in Fig. 1.

0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5

x 104

−5

−4

−3

−2

−1

0

1

2

3

4

5x 10

5

samples (1:50000)

ampl

itute

Example of a seismic signal

x(t)

t

Fig. 1. Seismic signal.

– A photograph image (2-D signal) is represented as a brightness function(s) of two spatial

variables (coordinates in a plane). As an example, Figure 2 shows 506 random balls placed randomlyin the 3-D box 640×520×100. At each point (x, y, z) of those balls, an intensity θ(x, y, z) is defined

that is concentrated in the center of balls (see Fig. 3).

Fig. 2. Image with random balls.

The projection of the intensity along the middle lines of Y-axis and X-axis are shown respectivelyon Figs. 4(a) and (b).

Signals convey information and include physical quantities such as voltage, current, and intensity.We will consider 1-D signals (analog or discrete) that are functions (continuous or step-wise) usually

of time or frequency. They are two important variables, time and frequency (t and ω), used in signalprocessing.


020

40

0

500

50

100

430

440

450

255

260

265

270

275

280

68

70

72

74

76

78

Slice of ball 58 on Z−axis 73

Fig. 3. Random balls.

0 100 200 300 400 500 600

50

100

150

200Intensity of balls (projection by y=256)

x

0 50 100 150 200 250 300 350 400 450 500

50

100

150

200Intensity of balls (projection by x=320)

y

Fig. 4. Projections of the intensity of random balls.

A. In the one-dimensional case, we used to consider the independent variable of the mathematical

representation of a signal as the time.The independent variable in the mathematical representation of a signal may be either continuous

or discrete.Continuous-time signals are often refereed to as analog signals and are defined along a continuum

of times.Discrete-time signals are refereed mathematically as a sequences of numbers and those are defined

at discrete times (the independent variable has discrete values).

Signals may have either a continuous or discrete variable representation, and if certain conditionshold, these representations are entirely equivalent.

We use also concepts of continuous and discrete signals respectively as those for which thefunctions (represented them) continuous or discrete.


B. Analog signals can be described by a mathematical expression or graphically by a curve, orby a set of tabulated values. Real signals are not easy to describe quantitatively. They must often

be approximated by idealized forms or models.Signals can be of finite or infinite duration. Finite durations signals are called time-limited.

Signals of semi-infinite extent may be right-side, if they are zero for t < a (a is finite), or left-side,if they are zero for t > a. Signals are causal if they are zero for t < 0.

C. Model of continuous and discrete signals (example of the MATLAB-based program):

0 50 100 150 200 250−2

−1

0

1

2Continuous function (model)

magnit

ude

y(t)=2exp(−αt)cos(βt)

t

0 20 40 60 80 100 120−2

−1

0

1

2Discrete sequence of data

Number of points

magnit

ude

t

a)

b)

Fig. 5. Graphic representations of sequences: (a) analog signal y(t) = 2 exp(−αt)sin(βt) for α = 0.02,

β = 0.25, and (b) discrete analog of the signal, y(n).


% Call: figure_5.m

% To calculate and display the signal function (as in Fig. 5)

%

% y(t)=2exp(-\alpha t)sin(\beta t)

%

% Dr. Art Grigoryan, EE 3423 UTSA May 30, 2002

% 2-D stem plot

alpha = .02; % input data

beta = .250;

N=1024; % number of points

t=0:1:N; % interval for time parameter

y=2*exp(-alpha*t).*sin(beta*t); % vector multiplication (.*)

%1 Model of the continuous function

figure;

subplot(2,1,1);

plot(t,y);

axis([0,N/4,-2,2]); % to display only 1/4 part of y

S_title1=sprintf(’Continuous function (model)’);

h_title1=title(S_title1);

set(h_title1,’Color’,[1 0 .3],’FontSize’,10,’FontName’, ’Times’);

h_tx1=text(90,1.2,’y(t)=2exp(-\alpha{t})cos(\beta{t})’);

set(h_tx1,’Color’,’b’,’FontName’,’Times’,’FontSize’,9);

h_y1=ylabel(’magnitude’);

set(h_y1,’Color’,’b’,’FontName’,’Times’,’FontSize’,9);

h_x1=text(230,-2.3,’t’);

set(h_x1,’Color’,’b’,’FontName’,’Times’,’FontSize’,9);

%2 Model of the discrete signal

t1=1:1:N;

y1=2.*exp(-alpha*t1).*sin(beta*t1);

subplot(2,1,2);

stem(t1,y1);

axis([0,N/8,-2,2]); % to display only 1/8 part of y

S_title2=sprintf(’Discrete sequence of data’);

h_title2=title(S_title2);

set(h_title2,’Color’,[1 0 .3], ’FontSize’, 10, ’FontName’, ’Times’);

h_y2=ylabel(’magnitude’);

set(h_y2,’Color’,’b’,’FontName’,’Times’,’FontSize’,9);

h_x2=xlabel(’number of points’);


h_x2=text(110,-2.3,’t’);


h_a=gtext(’a)’); h_b=gtext(’b)’);

set(h_a,’FontName’,’Times’,’FontSize’,10);

set(h_b,’FontName’,’Times’,’FontSize’,10);

% print -depsc fig_8.ps;


A. Transformations of signals

Let x(t) be a function of one independent variable (time) defined for all values −∞ < t < +∞,i.e. for all t ∈ R, where R = R1 denotes the real line.

A.1 Time transformations.

1. Time reversal. Given a signal x(t), a time-reversal transformation of the signal is defined as

y(t) = x(−t), −∞ < t < +∞. (1)

Drawing the graph of the signal, we can see that the time-reversal transformation creates the mirrorimage about the vertical axis.

As an example, Figure 7) shows the signal x(t) defined in the interval [−1, 2] in part a, alongwith the time-reversal transform in b and c, and two signals together in d. Note that in c, the

numbers along the x-axis have been changed only.

−2 −1 0 1 2−2

−1

0

1

2

3

(a)

mag

nit

ude

−2 −1 0 1 2−2

−1

0

1

2

3

(b)

2 1 0 −1 −2 −2

−1

0

1

2

3

(c)

mag

nit

ude

−2 −1 0 1 2−2

−1

0

1

2

3

(d)

x(t)

x(t)x(−t)

x(−t)

t

t

t

t

Fig. 7. Signal and its time-reversal transform.


There are many functions which do not change after time-reversal transformation, i.e.

y(t) = x(t), ∀t.

For example, such are signals x(t) = |t|, x(t) = t2, x(t) = cos(x), which are called symmetric relative

the vertical axis t = 0 These symmetric signals are shown in Fig. 8 along with non symmetricfunctions t3 and sin(t).

−2 −1 0 1 2−4

−3

−2

−1

0

1

2

3

4

t2

|t|

t3

symmetric functions

(a)

t2

t2

t2

−5 0 5−1.5

−1

−0.5

0

0.5

1

1.5x(t)=cos(t)

(b)

x(t) x(t)

t t

Fig. 8. Symmetric and non symmetric signals.

Note that the above signal x(t) is defined only on the interval [−1, 2]. We can thus define and

plot y(t) only on the interval [−2, 1], i.e. y(t) is defined by (1) when t ∈ [−2, 1].In practice, working with digital signals which are defined (or can be defined) on a finite interval

[a, b], we used to determine the time-reversal transformation of the signal relative to the verticalline crossing the middle point of [a, b]. In order words, we define the time-reversal transformationsas (See Fig. 9)

y(t) = x(b + a − t), t ∈ [a, b]. (2)

That results in the mirror image about the vertical axis shifted by (b + a)/2 to the right. Indeed,

we can write that

x(b + a − t) = x

(

b + a

2−

[

t −b + a

2

])

.

0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5−2

−1

0

1

2

3

mag

nit

ude

t0=(a+b)/2

a=1, b=4

window

x(t) y(t)

Fig. 9. Time-reversal signals relative to the middle point.

We face here with a new operation over the function, namely the time-shifting of the signal.


2. Time shifting. Given a signal x(t) and a value t0, a time-shifted version of the signal is definedas

y(t) = xt0(t) = x(t − t0), −∞ < t < +∞. (3)

When we plot the signal, the time-shifting transformations yields the shift of the vertical axis byvalue t0 to the left or right, depending on t0 > 0 or t0 < 0.

As an example, Figure 10 shows the signal x(t) and its shifted transforms by ±3.5

y(t) = x(t − 3.5) (shifted to the right by 3.5)

and

y(t) = x(t + 3.5) (shifted to the left by 3.5).

−5 −4 −3 −2 −1 0 1 2 3 4 5 6−2

−1

0

1

2

3

mag

nit

ud

e

shifted signals

x(t)

x(t+3.5) x(t−3.5)

t

Fig. 10. Signal x(t) and its time-shift transforms by ±3.5.

Example 1: We consider the cosine signal

x(t) = cos(t), t ∈ R.

Taking values of t0 equal π/2 and −π/2, we obtain respectively (See Fig. 11):

xπ/2(t) = x(t − π/2) = cos(t − π/2) = − sin(t)

x−π/2(t) = x(t + π/2) = cos(t + π/2) = sin(t).

−8 −6 −4 −2 0 2 4 6 8−1.5

−1

−0.5

0

0.5

1

1.5shifted signals

mag

nit

ude

x(t)=cos(t)

xπ/2

(t)x−π/2

(t)

Fig. 11. The signal x(t) = cos(t).


Example 2: We consider the following signal, which plays important role in digital signal pro-cessing,

x(t) = e−t, t ≥ 0, x(t) = 0, t < 0.

The shift by t0 yields the following amplification of the signal

xt0(t) = x(t − t0) = e−(t−t0) = et0e−t = ae−t (4)

for t ≥ t0, and xt0(t) = 0, otherwise. The amplitude of the time shifted signal at every point of twill increase or decrease for t ≥ t0 depending on a > 1 or a < 1, respectively (a = et0) (See Fig. 12)

0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 50

0.5

1

1.5

2

2.5

signal

mag

nit

ud

e

Time in µsecs

x1(t)=exp(−(t−1))

x(t)=exp(−t)

x−1

(t)=exp(−(t+1))

t

Fig. 12. The shift of the signal x(t) = e−t, t ≥ 0.

Task 1: For signal which is described as

x(t) = e−t cos(t), t ≥ 0, x(t) = 0, t < 0,

write and plot the time-shifted signals for t0 = π/2, −π/2, π, π/4, and −π/4.

3. Time scaling. Given a signal x(t) and a positive constant a, a time scaled version of the signalis defined as

y(t) = x(at), −∞ < t < +∞. (5)

As an example, Fig. 13 shows the time scaling transformations for a = 2 and a = 1/2. The signalcan be defined on the whole line R = (−∞, +∞) which does not change after multiplying by a 6= 0,i.e. aR = R.

Figure 14 shows the time-scaling of the cosine wave, when the scale factors are 2 and 1/2.


−2 −1 0 1 2−2

−1

0

1

2

3

(a)

mag

nit

ud

e

−2 −1 0 1 2−2

−1

0

1

2

3

(b)

−2 −1 0 1 2−2

−1

0

1

2

3

(c)

mag

nit

ude

−2 0 2 4−2

−1

0

1

2

3

(d)

x(t)

x(t)x(t/2)

x(2t)

t

t

t

t

Fig. 13. (a) Signal x(t) and its time-scaling versions for (b) a = 2, (c) a = 1/2, and (d) all signals together.

−6 −4 −2 0 2 4 6

−1

0

1

Time−transformation of the signal

mag

nit

ude

x(t)=cos(t)

−6 −4 −2 0 2 4 6

−1

0

1

mag

nit

ude

y(t)=cos(2t)

−6 −4 −2 0 2 4 6

−1

0

1

mag

nit

ude

y(t)=cos(t/2)

t

Fig. 14. Cosinusoudal signal x(t) = cos(t) and time-scaled versions y(t) = cos(2t) and y(t) = cos(t/2).

3. Combination of time transformations.

We consider the general time transformation, which is defined as

y(t) = x(at − t0), ∞ < t < +∞, (6)


where a > 0 is a value of time scaling, t0 is a value of time shifting. Denoting the new variable

t′ = at − t0

we can write thaty(t) = x(t′), t = (t′ + t0)/a. (7)

Example 3: Let a = 1/2 and t0 = 1,

y(t) = x

(

1 −1

2t

)

= x(t′)

t = 2 − 2t′

The construction of y(t) is shown in Fig. 15(c).

We can also plot y(t) by using the following step-by-step transformations:

x′(t) = x(−t) (8)

x′

1/2(t) = x′(t/2) (9)

x′

1/2(t − 2) = x′([t − 2]/2) = x′(t/2 − 1) (10)

= x(1− t/2) (11)

which is illustrated in Fig. 15.

−2 −1 0 1 2−2

−1

0

1

2

3

(a)

mag

nit

ude

−2 −1 0 1 2−2

−1

0

1

2

3

(b)

−2 0 2 4−2

−1

0

1

2

3

(c)

mag

nit

ude

−4 −2 0 2−2

−1

0

1

2

3

(d)

x(t)

x(−t/2)y(t)

x(−t)

t

t

t

t

Fig. 15. Construction of the time-transformation y(t) of the signal x(t) in Example 3.

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