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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,
𝑥 𝑛 = … ,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
seconds, then the sampling frequency is given by 𝑓𝑠 =1
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.
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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.
𝑢 𝑛 = 0 𝑛 < 01 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:
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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.
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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) Real vs. Complex Signals A sequence is considered complex at
least one sample is complex-valued.
ii) 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|.
iii) Causal Signals A sequence x[n] is a causal sequence if x[n]
= 0 for n < 0.
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iv) Symmetric Signals First consider a real-valued sequence
{x[n]}.
Even symmetry implies that x[n] = x[-n] and Odd symmetry implies
that x[n] = -x[n] for all n.
Any real-valued sequence can be decomposed into odd and even
parts so that
where the even part is given by
and the odd part is given by
A complex-valued sequence is conjugate symmetric if x[n] =
x*[-n]. The sequence has
conjugate anti-symmetry if x[n] = -x*[-n]. Analogous to
real-valued sequences, any complex-valued sequence can be
decomposed into its conjugate symmetric and conjugate anti-
symmetric parts:
v) Periodic Signals
A discrete-time sequence is periodic with a period of N samples
if
for all integer values of k. Note that N has to be a positive
integer.
If a sequence is not periodic, it is aperiodic or
non-periodic.
Consider a discrete-time sequence x [n] based on a sinusoid with
angular frequency ωo:
If this sequence is periodic with a period of N samples, then
the following must be true:
However, the left hand side can be expressed as
and the cosine function is periodic with a period of 2π and
therefore the right hand side
of the above equation is given by
for integer values of r.
Comparing the above two equations, we have
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where ωo = 2πfo. Since both r and N are integers, a
discrete-time sinusoidal sequence is
periodic if its frequency is a rational number. Otherwise, it is
non-periodic.
vi) Energy and Power Signals The energy of a finite length
sequence x[n] is defined as
while that for an infinite sequence is
Note that the energy of an infinite length sequence may not be
infinite. A signal with finite
energy is usually referred to as an energy signal.
The average power of a periodic sequence with a period of N
samples is defined as
and for non-periodic sequences, it is defined in terms of the
following limit if it exists:
A signal with finite average power is called a power signal.
vii) Bounded Signals A sequence is bounded if every sample of
the sequence has a magnitude which is less than or
equal to a finite positive value. That is,
viii) Summable Signals A sequence is absolutely summable if the
sum of the absolute value of all its samples is finite.
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A sequence is square summable if the sum of the magnitude
squared of all its samples is
finite.
Classification of Systems
Discrete-time systems, like continuous-time systems, can be
classified in a variety of
ways.
i) Linearity A linear system is one which obeys the
superposition principle. For a certain system, let the outputs
corresponding to inputs x1[n] and x2[n] are y1[n]
and y2[n] respectively. Now if the input is given by
where A and B are arbitrary constants, then the system is linear
if its corresponding output is
Superposition is a very nice property which makes analysis much
simpler. Linearization is a
very useful approach to analyzing nonlinear systems.
ii) Shift Invariance A shift (or time) invariant system is one
that does not change with time. Let a system
response to an input x[n] be y[n]. If the input is now shifted
by n0 (an integer) samples,
then the system is shift invariant if its response to x1[n]
is
We can use the terms linear time-invariant (LTI) and linear
shift-invariant interchangeably.
iii) Causality The response of a causal system at any time
depends only on the input at the current and past
instants, not on any future samples. In other words, the output
sample y[no] for any no only
depends on x[n] for n ≤ n0.
iv) Stability There are two common criteria for system
stability. They are exponential stability and
bounded-input bounded-output (BIBO) stability. It requires the
response of the system to
decay exponentially fast for a finite duration input. The second
one merely requires that the
output be a bounded sequence if the input is a bounded
sequence.
Linear Shift-Invariant Systems
A discrete-time LTI system, now called as LSI system, like its
continuous-time
counterpart, is completely characterized by its impulse
response. In other words, the impulse
response tells us everything we need to know about an LSI system
as far as signal processing
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is concerned. The impulse response is simply the observed system
output when the input is an
impulse sequence. The output of the LSI system is using
Convolution sum.
Condition for Stability
Since the impulse response completely characterizes an LSI
system, we draw conclusions
regarding the stability of a system based on its impulse
response.
Theorem 1.1.
An LSI system is BIBO stable if its impulse response is
absolutely summable.
Proof: Let the input be bounded, i.e. |x[n]| < B < 1 for
some finite value B. The magnitude of the
output is given by
So the magnitude of y[n] is bounded if is finite. In other
words, the impulse
response must be absolutely summable.
Condition for Causality
Theorem 1.2.
An LSI system is causal if and only if its impulse response is a
causal sequence.
Proof. Consider an LSI system with impulse response h[k]. Two
different inputs x1[n] and x2[n] are
the same up to a certain point in time, that is x1[n] = x2[n]
for n ≤ n0 for some no. The outputs
y1[n] and y2[n] at n = no are given by
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and
Since x1[n] = x2[n] for n ≤ no, if the system is causal, then
the outputs y1[n] and y2[n] must be
the same for n ≤ no. More specifically, y1[no] = y2[no].
Now,
because for non-negative values of k. Since x1[n] may not be
equal to x2[n] for n > no, we must have
which means that h[k] = 0 for k < 0.
Illustrative Examples:
Problem 1: Check whether the system defined by y(n) = x2(n) is
shift-invariant.
Solution:
If y(n) = x2(n) is the response of the system to x(n), the
response of the system to
x'(n) = x(n - no) is
Because y'(n) = y(n - no), the system is shift-invariant.
Problem 2: Check whether the systems described by the
equations
i) y(n) = x(n) +x(n - 1 ) ii) y(n) = x(n) + x(n+1)
Solution:
i) y (n) = x(n) +x(n - 1 ) is causal because the value of the
output at any time n = no depends only on the input x(n) at time no
and at time no - 1.
ii) y(n) = x(n) + x(n+1) is noncausal because the output at time
n = no depends on the value of the input at time no + 1.
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Problem 3: An LSI system with unit sample response h(n) = anu(n)
will be stable whenever
|a| < 1. Justify.
Solution:
We have learnt that the magnitude of y[n] is bounded if is
finite i.e., the
impulse response must be absolutely summable.
Hence, the impulse response is absolutely summable only when |a|
< 1
Problem 4: Prove that the system described by the equation y(n)
= n x(n) is not BIBO stable.
Solution:
Assume that the input signal is a unit step signal x(n) = u(n)
which is bounded.
𝑢 𝑛 = 0 𝑛 < 01 𝑛 ≥ 0
But the response to a unit step is y(n) = n u(n), which is
unbounded as n increases
Hence the system is not BIBO stable.
Problem 5: Find the response of the system with impulse response
h(n) = u(n) for the input
x(n)=anu(n).
Solution:
With the direct evaluation of the convolution sum we find the
response of the system
Because u(k) is equal to zero for k < 0 and u(n - k) is equal
to zero for k > n, when n < 0,
there are no nonzero terms in the sum and y(n) = 0. On the other
hand, if n ≥ 0,
Since
Therefore,
Summary:
Discrete-time signals are essentially a sequence of numbers.
Some fundamental
sequences, such as the unit impulse, unit step and sinusoidal
(both real and complex)
sequences are examined because more complex sequences can be
expressed as a combination
of some or all of these fundamental ones.
The most important discrete-time systems are the linear shift
invariant (LSI) systems.
For these systems, the superposition principle applies which
leads to the linear convolution
sum. This convolution sum is the way by which we derive the
output of the system given an
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input. An LSI system is completely characterized by its impulse
response. All the properties
of such a system is revealed by its impulse response, including
causality and BIBO stability.
Assignment:
Problem 1: Check whether the system described by the equation is
shift- varying
y(n)=x(n)+x(-n)
Problem 2: The input x[n] and output y[n] of a system are linked
by the relation
y[n] = T{x[n]} = x[- n].
Answer the following questions, giving reasons: a) Is the system
linear? (b) Is the system
time invariant? c) Is the system causal?
Problem 3: Determine which of the following systems are
stable:
a) y(n) = ex(n)
/ x(n - 1)
b) y(n)=cos(x(n))
Problem 4: A linear shift-invariant system has a unit sample
response h(n)=u(-n-1)
Find the output if the input is x(n) =3nu(-n)
Problem 5: The input to a linear shift-invariant system is the
unit step, x(n) = u(n), and the
response is y(n) = δ(n). Find the unit sample response of this
system.
Simulation:
Convolution is a formal mathematical operation, just as
multiplication, addition, and
integration. Addition takes two numbers and produces a third
number, while convolution
takes two signals and produces a third signal. Convolution is
used in the mathematics of
many fields, such as probability and statistics.
A convolution of two functions is denoted with the operator “
*”, and is written as
Convolution Sum (Discrete functions)
Program:
clc; clear all; close all;
x=input('Enter the input'); %Taking input1 from the user
h=input('Enter the impulse response'); %Taking input2 from the
user
l=length(x); %Finding lengths m=length(h);
subplot(3,1,1); %plotting the input sequences stem(0:l-1,x);
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xlabel('n---->'); ylabel('x(n)--->'); title('Input
sequence'); subplot(3,1,2); stem(0:m-1,h); xlabel('n---->');
ylabel('h(n)--->'); title('Impulse response sequence');
%convolution logic y=[]; for n=2:l+m y1=0; for k=1:max(l,m) if
l>=m && n-k>0 && n-kl && n-k>0
&& n-k
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References:
1. Digital Signal Processing, Principles, Algorithms and
Applications – John G Proakis, Dimitris G Manolakis,
Pearson Education / PHI, 2007
2. Discrete Time Signal Processing – A V Oppenheim and R W
Schaffer, PHI, 2009
3. Digital Signal Processing – Monson H.Hayes – Schaum’s
Outlines, McGraw-Hill,1999
4. Fundamentals of Digital Signal Processing using Matlab –
Robert J Schilling, Sandra L Harris, Thomson
2007.
5. Digital Signal processing – A Practical Approach, Emmanuel C
Ifeachor and Barrie W Jervis, 2nd
Edition, PE
2009
6. Digital Signal Processing – A Computer Based Approach, Sanjit
K.Mitra, McGraw Hill,2nd
Edition, 2001