Digital Signal Processing

Lab#1

Abstract— understanding the concept of continuous-time and

discrete-time signals and systems allows engineers to analyze and

to employ those signals through acquisition, representation,

manipulation, and transformation in a wide range of practical

applications.

Key Terms—sample, amplitude, periodic signal, pulse and step

functions, complex signal, sequence, MATLAB, convolution,

continuous signal, discrete signal, digital signal.

I. INTRODUCTION

Signals are physical quantities that transmit information in

their patterns of deviation. To simplify the analysis and design

of signal processing systems is vital to represent signals by

mathematical functions of one or more independent variables.

The essential mathematical aspects of signal processing deal

with ideal discrete-time signal processing systems, and ideal

A/D and D/A converters. Digital signal processing involves

the conversion of analog signals into digital, processing the

obtained sequence of finite precision numbers using a digital

signal processor or general purpose computer, and if

necessary, converting the resulting signal sequence back into

analog form. The most widely used for analysis and design is

MATLAB, a numerical computing environment. MATLAB

uses the a computer algebra system symbolic engine for

graphical multi-domain simulation and model-based design of

dynamic and embedded systems.

II. GENERATION DATA USING MATLAB

A. Unit amplitude rectangular window for 51 samples:

The rectangular window with unit amplitude may be

defined by

[ ] {

(1)

where M is the window length in samples. A plot of the

rectangular window appears in Fig.1 for length M=25.

B. Square-wave with 20 samples per cycle:

The square wave with amplitude ±1 may be defined by

Fig.1. The rectangular window.

[ ] { (

)

(2)

where k is the number of samples per half of sinusoid

period. A plot of the square wave appears in Fig.2 with 10

samples ON and 10 samples OFF; total of three cycles.

C. Delta function:

The delta function that represents impact of the unit-

impulse on some signal x[n] is represented by Eq.3.

Fig.2. The square wave.

[ ] [ ] [ ] [ ] (3)

where 𝞭[n] is the unit-impulse that may be defined by

[ ] {

. (4)

A plot of the delta function with the impulses at n=0, n= 30,

n= 90 appears in Fig. 3.

D. Sinusoid with 40 samples per cycle:

The sinusoid function is built by

[ ]

(5)

where n is the number of sample from some sampling

range,

represents how many samples is taken per period,

and

is a phase shift. A plot of the sinusoid appears in Fig. 4.

Fig. 3. The delta function.

Fig. 4. The sinusoid.

E. Complex exponential:

The complex exponential sequence

[ ] ( )

(6)

may be defined by

[ ] (7)

where A is an amplitude, is an angular frequency of

x[n]. A plot of the real and imaginary exponential sequence

appears in Fig. 5.

F. Triangle sequence:

The triangular pulse with 10 samples may be represented by

[ ] | (

) | (8)

and a plot of the triangular pulse of three cycles appears in

Fig. 6.

Fig. 5. The complex exponential sequence.

Fig. 6. The triangle pulse.

G. Hamming, Blackman, Hanning:

A window function is a mathematical function that is zero-

valued outside of some chosen interval. The Hamming

window may be defined by

[ ]

(9)

where M is chosen interval (51 samples). A plot of the

Hamming window appears in Fig. 7.

The Blackman window may be defined by

[ ] (

)

(10)

where M is chosen interval (51 samples). A plot of the

Blackman window appears in Fig. 8.

The Hanning window may be defined by

[ ] (

) (11)

where M is chosen interval (51 samples). A plot of the

Hanning window appears in Fig. 9.

Fig. 7. Humming window.

Fig. 8. The Blackman window.

Fig. 9. The Hanning window.

III. ANALYSIS OF SIGNALS

A. Family of continuous-time sinusoids:

The continuous-time sequence is a sequence whose value

y(t) is defined for every value of the independent variable t

(time). The general representation of such sequence is denoted

by Eq.12.

(12)

where A is amplitude, 𝜴 – angular frequency (rad/sec), θ –

phase shift (rad). A plot of the family of the continuous-time

sinusoids at frequencies 𝜴 π 𝜴 π 𝜴 π 𝜴 π A a d θ π app ar F g

B. Family of discrete-time sinusoids:

The discrete-time sequence is a sequence whose value y[n]

is defined only at a discrete set of values of the independent

variable n (usually the set of integers). The general

representation of such sequence is denoted by Eq.13.

(13)

where A is amplitude, – angular frequency (rad/sample),

–phase shift (real constant). A plot of the family of the

discrete-time sinusoids at frequencies =0, =0.1π,

=0.2π, =0.4π, =1π, =1.1π, =1.2π appears in Fig.11.

The Fig.11 represents Bizarre property #2 according to

which, frequencies in the neighborhood of ω=0 or 2πk are

called low frequencies, whereas, frequencies in the

neighborhood of ω=π 0r π(2k+1) are called high frequencies.

C. Convolution:

Convolution is a mathematical way of combining two signals

to form a third signal. Using the strategy of impulse

decomposition, systems are described by a signal called

the impulse response. Convolution is important because it

relates the three signals of interest: the input signal, the output

signal, and the impulse response.

The implementation of such technic is represented at

Fig.12.a trough Fig.12.f.

Fig. 10.The family of continuous-time sinusoids.

Fig. 11.The family of discrete-time sinusoids.

Fig. 12.a. The convolution of rectwin*triangle.

Fig. 12.b. The convolution of square*sinusoid.

Fig. 12.c. The convolution of compexp*square.

Fig. 12.d. The convolution of delta*hamming.

Fig. 12.e. The convolution of delta*compexp.

Fig. 12.f. The convolution of delta*square.

D. Convolution properties:

From a mathematical viewpoint the roles of h[n] and x[n]

in the convolution sum are equivalent. However, in the context

of linear time-invariant systems, the roles played by the

impulse response and the input are not equivalent. The nature

of h[n] determines the effect of the system on the input signal

x[n]. Since all linear time-invariant systems are described by a

convolution sum, Fig.13.a through Fig.13e represents the

properties of convolution to study their properties and

determine the impulse response of interconnected systems.

Fig. 13.a.The identity property x1[n] ∗ δ[n] = x1[n].

Fig. 13.b.The delay property x1[n] ∗ δ[n − n0] = x1[n − n0].

Fig. 13.c.The commutative property x1[n] ∗ x2[n]= x2[n] ∗ x1[n].

Fig. 13.d.The associative property

(x1[n] ∗ h[n]) ∗x2[n] = h[n]∗ (x1[n] + x2[n]).

Fig. 13.e.The associative property

h[n] ∗ (x1[n] + x2[n] ) = h[n]* x1[n] + h[n]* x2[n].

E. Answers to the book problems:

Problem #2.35

(a) y(t) = x(t − 1) + x(2 − t)

Answer: linear, time-invariant, noncausal, stable.

(b) y(t) = dx(t)/dt

Answer: linear, time-invariant, causal, stable.

(c) y(t) = ∫

Answer: linear, time-varying, noncausal, unstable.

(d) y(t) = 2x(t) + 5.

Answer: nonlinear, time-invariant, causal, stable.

The solutions are presented in the Appendix A.

Problem #2.40

y[n] = 10x[n] cos(0.25πn + θ)

Answer: linear, time-varying, causal, stable.

F. Discrete-time system input-output relation:

Given the discrete-time system is defined by

[ ] [ ] [ ]

[ ] . (14)

For the input x[n]=au[n] with y[-1]=1 as n→∞, where a is

a positive number, the system represents square root of the a.

The system is time-invariant because of

[ ] [ ] [ ]

[ ]. (15)

The test for linearity states that the system is nonlinear; a

solution is presented in the Appendix B.

A plot of the system input and output relation appears at

Fig.14.

Fig. 14.The input-output relation of [ ] [ ] [ ]

[ ] .

IV. APPENDIX

A. Solutions for the book problems.

B. Testing the system [ ] [ ] [ ]

[ ] for linearity.