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AAR MAHAVEER ENGINEERING COLLEGE
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Digital Signal Processing
Module 1
Analysis of Discrete time Linear Time - Invariant Systems
Objective:
1. To understand the representation of Discrete time signals
2. To analyze the causality and stability concepts of Linear Shift Invariant (LSI) systems
Introduction:
Digital signals are discrete in both time (the independent variable) and amplitude (the
dependent variable). Signals that are discrete in time but continuous in amplitude are referred
to as discrete-time signals.
A discrete-time system is one that processes a discrete-time input sequence to produce
a discrete-time output sequence. There are many different kinds of such systems. One of the
important kinds is Linear Shift Invariant (LSI) systems.
Description:
Discrete time Signals and Sequences
Discrete-time signals are data sequences. A sequence of data is denoted {x[n]} or
simply x[n] when the meaning is clear. The elements of the sequence are called samples. The
index n associated with each sample is an integer. If appropriate, the range of n will be
specified. Quite often, we are interested in identifying the sample where n = 0. This is done
by putting an arrow under that sample. For instance,
𝑥𝑛
= … ,0.35,1,1.5, −0.6, −2, …
The arrow is often omitted if it is clear from the context which sample is x[0]. Sample values
can either be real or complex. The terms “discrete-time signals” and “sequences” are used
interchangeably.
The time interval between samples is can be assumed to be normalized to 1 unit of time. So
the corresponding normalized sampling frequency is 1Hz. If the actual sampling interval is T
1
seconds, then the sampling frequency is given by 𝑓𝑠 = 𝑇
Some Elementary Sequences
i)
Unit Impulse Sequence
The unit impulse sequence is defined by
This is depicted graphically in Figure 1.1. Note that while the continuous-time unit impulse
function is a mathematical object that cannot be physically realized, the unit impulse
sequence can easily be generated.

Figure 1.1: The Unit Impulse Sequence
ii)
Unit Step Sequence
The unit step sequence is one that has amplitude of zero for negative indices and amplitude of
one for non-negative indices.
0 𝑛<0
𝑢 𝑛 =
1 𝑛≥0
It is shown in Figure 1.2.
Figure 1.2: The Unit Step Sequence
iii)
Exponential sequences
The general form is
If A and α are real numbers then the sequence is real. If 0 < α < 1 and A is
positive, then the sequence values are positive and decrease with increasing n:

iv)
Sinusoidal Sequences
A sinusoidal sequence has the form
This function can also be decomposed into its in-phase xi[q] and quadrature xq[n]
components.
This is a common practice in communications signal processing. It is shown in Figure 1.3.
Figure 1.3: The Sinusoidal Sequence
v)
Complex Exponential Sequences
Complex exponential sequences are essentially complex sinusoids.
vi)
Random Sequences
The sample values of a random sequence are randomly drawn from a certain probability
distribution. They are also called stochastic sequences. The two most common
distributions are the Gaussian (normal) distribution and the uniform distribution. The
zero-mean Gaussian distribution is often used to model noise. Figure 1.4 and Figure 1.5
show examples of uniformly distributed and Gaussian distributed random sequences
respectively.

Figure 1.4: Uniformly distributed random sequence with amplitudes between -0.5 and 0.5.
Figure 1.5: Gaussian distributed random sequence with zero mean and unit variance.
Types of Sequences
The discrete-time signals that we encounter can be classified in a number of ways. Some
basic classifications that are of interest to us are described below.
i)
ii)
iii)
Real vs. Complex Signals
A sequence is considered complex at least one sample is complex-valued.
Finite vs. Infinite Length Signals
Finite length sequences are defined only for a range of indices, say N1 toN2. The
length of this finite length sequence is given by |N2- N1 + 1|.
Causal Signals
A sequence x[n] is a causal sequence if x[n] = 0 for n < 0.

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