- Digital Signal Processing (DSP) involves discrete operations on signals like shifting, flipping, and scaling. - Signals can be decomposed into even and odd parts using transformations. Periodic signals repeat after a certain interval. - Important signals include the delta function, unit step function, and their relationship. Signals can be represented in polar or Cartesian coordinates. - Real exponential and complex exponential signals are discussed. In discrete time, the lowest frequency is 1 and highest is the fastest possible digitization rate. - Ideal filters are aimed for but practical filters may differ. Periodicity of signals needs careful consideration in discrete time.

1. Digital Signal Processing
Lecture 1 Contents (Signals)
Follows Section 2.1 of the textbook (Proakis and
Manolakis, 4th ed.).
By
Dr. Muhammad Imran Farid

2. • What is a signal? What is a system?
• Continuous time vs. discrete time (analog vs. digital)
• Signal transformations
• Flipping/time reversal
• Scaling
• Shifting
• Combining transformations; order of operations
• Signal properties
• Even and odd
• Decomposing a signal into even and odd parts (with Matlab demo)
Lecture 1 Contents

3. • Periodicity
• Special signals
• The delta function
• The unit step function
• The relationship between the delta and step functions
• Decomposing a signal into delta functions
• The sampling property of delta functions
• Complex number review (magnitude, phase, Euler's formula)
• Real sinusoids (amplitude, frequency, phase)
• Real exponential signals
• Complex exponential signals
Lecture 1 Contents

4. Digital Signal Processing
Lecture 1 (Signals)
By
Dr. Muhammad Imran Farid

5. • Digital Signal Processing (DSP) is a discrete version of Signal and Systems

7. • Basic Operations to a signal

8. Verify by plugging some numer

9. shift
flip
scale
to check
Check by plugin values

11. % code for even and odd in DSP Lecture 1
clear all;
close all;
clc;
x = rand(1,9) - 0.5;
figure(1), stem(x,'LineWidth',2);
% to make the middle value zero
figure(2), stem(-4:4, x,'LineWidth',2);
% flip version of x
negx = fliplr(x); % flip left to right
figure(3), stem(-4:4, negx,'LineWidth',2);
% even and odd part of x
evx = (x + negx)/2;
odx = (x - negx)/2;
figure(4), stem(-4:4, evx,'LineWidth',2);
figure(5), stem(-4:4, odx,'LineWidth',2);
% verify whether we get beck original signal from even odd or nor ?
q = evx + odx;
figure(6), stem(-4:4, q,'LineWidth',2);

12. (signal repeats after certain integer)

13. Delta function
Unit Step
function

14. • An another way of seeing the relationship between continuous and discrete delta
function

15. Continuous time
step function
derivative
Continuous Time
delta function
Life is easier in
digital world
Delta function is just the
difference between the
two step functions
Analogy to continuous part

16. • For every value of k of delta, we are multiplying it with the corresponding values of X and add them all
up to get X[n]
• We use it a lot in convolution etc….

17. • Lets say we have a signal as a function of
k
• We can pick up any value of signal using delta function

19. Cartesian to
polar
Polar to
Cartesian

24. Real envelope on
top of sinusoid
Not periodic (because amplitude change) but a sense of
periodicity inside the envelope
• If r < 0 it’s the decreasing envelope
• If r = 0 we don’t have envelope and it’s the regular sine and cosine
• If r > 0 it’s the increasing envelope

25. • In a similar way in discrete time signal

26. • In a similar way in discrete time signal
• If 𝛽 < 0 it’s the decreasing envelope
• If 𝛽 = 0 we don’t have envelope and it’s the regular sine and cosine
• If 𝛽 > 0 it’s the increasing envelope
𝛽 < 0

28. • If we add 2𝜋 in any frequency, we ends up getting the same frequency
• Means there is no infinitely high frequencies in discrete time world

29. The lowest frequency one can get in
discrete world is one
• The highest frequency one can get
in discrete world is the back and
forth as quickly as possible
• This is the as fast digitally we can
go

30. Ideal
in practice
• Aim is to design a filter as close as possible to the ideal

31. • We need to be careful to determine whether a signal is periodic in the discrete world.
• Cosine is periodic but that is not always true in discrete world
Period looks like this

32. EXAMPLE:
for N to be an integer we must
have k = 2 for N to be 5
This will never be an integer

33. % code for even and odd in DSP Lecture 1
clear all;
close all;
clc;
n = -10:10;
x = cos(4*pi/5*n);
figure(1), stem(x,'LineWidth',2);
%non periodic example
x = cos(7*n);
figure(2), stem(x,'LineWidth',2);
• So we need to be very careful about very particular cosine that are periodic in discrete time world
• We will talk a lot about it in when Insha'Allah we will cover Fourier Series and Fourier Transform