Discrete time signals and systems
Chapter 2 (Part 1)
Outline
• Articles:
2.1 : 2.1.1 – 2.1.3
2.2 : 2.2.1 – 2.2.4
2.3 : 2.3.1 – 2.3.5 and 2.3.7
• Examples: All examples from 2.1.1 till
2.3.5except 2.3.6 and 2.3.7
• Problems: 2.2, 2.6, 2.7 (a,b,g,I,j), 2.10, 2.11, 2.16
(a, b(2, 4, 9), 2.19, 2.20, 2.22 (MATLAB)
Discrete time signals
• Signal is a function of the independent variable n called
sample
• The instances between sample are not defined and not zero
• There are various representations of discrete signals. One
of the representations is the sequence representation:


x ( n)  3,4, 5,8
x ( n)  3,4,5,8, 
x ( n)   ,3,4, 5,8, 



infinite
finite (4 point)
all zeros before 3
Basic Signals
• Unit sample or impulse:
1, for n  0
 ( n)  
0, for n  0
Basic Signals contd.
• Unit step:
1, for n  0
u ( n)  
0, for n  0
Basic Signals contd.
• Unit ramp:
n, for n  0
u r ( n)  
0, for n  0
Basic Signals contd.
• Real exponential signal:
x(n)  a n for n  0
Basic Signals contd.
• Complex exponential signal:
when,
a  re j 2f  
then the complex exponentia l signal becomes :
x(n)  r n cos2f   n  j sin 2f   n
such that,
xR (n)  r n cos2f   n
xI (n)  r n sin 2f   n
and we can plot them both seperately
Also,
x(n)  A(n)  r n
is the magnitude function of the complex signal
Angx(n)   (n)  2f   n
is the phase function of the complex signal
n j ( 2 (1/ 20)) n
x
(
n
)

(
0
.
9
)
e
Complex exponential:
n j ( 2 (1/ 20)  / 3) n
x
(
n
)

(
0
.
9
)
e
Complex exponential:
Classification of Discrete-Time Signals
Energy and Power signals:
• The energy of a signal is defined as: E 


x ( n)
2
n  
• It means that we have to take the magnitude of each sample and
square it then sum all these values.
• If E is finite, then the signals is called an energy signal.
• Many signals have infinite energy as well.
• The average power is defined as:
N
1
2
P  lim
x
(
n
)

N  2 N  1
n N
But
EN 
N

n N
x ( n)
2
is the energy of the signals from the interval  N  n  N
• Hence, the energy of a signal can be expressed as:
E  lim E N
N 
And the average power as,
1
P  lim
EN
N  2 N  1
• Note: If E is finite, the P = 0. If E is infinite, the average
power P may be finite or infinite.
• If P is finite but not zero, then the signal is called power
signal.
• Periodic signals are power signals.
• Example 2.1.1
Classification of Discrete-Time Signals contd.
Even and odd signals:
• If x(n)  x(n) then it is called even and if x( n)   x(n) then it
is called odd.
• Any signal is represented by the sum of the even and odd
components of a signal.
x(n)  xe (n)  xo (n)
such that,
1
xe (n)  x(n)  x(n)
2
1
xo (n)  x(n)  x( n)
2
Original Signal
10
5
0
-5
-6
-4
-2
0
2
4
6
2
4
6
2
4
6
2
4
6
Even Component
10
5
0
-6
-4
-2
0
Odd Component
5
0
-5
-6
-4
-2
0
Even + Odd = Original Signal
10
5
0
-5
-6
-4
-2
0
Operations on Signals
Delay and Advancement:
• Delaying the signal by k means shifting the signal k samples to
the right e.g. x(n-3) and it is written as TDk x(n)  x(n  k )
• Advancing the signal by k means shifting the signal k samples to
the left e.g. x(n+3).
• Folding means x(-n) and it is written as FDx(n)  x(n)
Note:
• x(n-k) means delay x(n)
• x(-n-k) means advance x(-n)
x(n+k) means advance x(n)
x(-n+k) means delay x(-n)
Time scaling
• If x(n) is a discrete signal, then :
y (n)  x(an) is called time scaling where a is integer
• This operation is called down sampling because the resultant
signal y(n) will miss some samples from the original signal x(n).
• Note that a can not be less than 1.
• If the samples are taken from an analog signal xa (t ) at a sampling
duration of T sec, then x(n)  xa (nT ) . Now if y (n)  x(an)
then, y (n)  xa (anT ) or the sampling duration becomes aT
instead of T. This is why this operation is called down sampling.
Example 2.1.4
Down sampling on analog signals
Addition, Multiplication and Sequence scaling
• Addition is done point wise: y(n)  x1 (n)  x2 (n)
• Multiplication is done point wise: y(n)  x1 (n) x2 (n)
• Sequence scaling:
y (n)  Ax(n)
if A  1  Amplificat ion and
if A  1  Attanuatio n
• Sequence scaling does not result in change in frequency or
phase. Only the power is changed.
Extra problems
1- If,
2n  1, for  3  n  3
x ( n)  
otherwise
0,
Then find and plot y (n)  0.5 x(2n  1) . Also discuss the
different operation performed on the signal x(n) which results in y(n).
2- The analog signal given below is sampled at a rate of 50samples/s
which results in x(n). Now what is the required sampling rate to
produce a signal y(n) = x(3n). Does aliasing occurs in case of y(n).
What is going to be the sampling frequency required to up-sample
the analog signal by a factor of 4 from the original sampling rate of
50Hz. Plot all signals in MATLAB. xa (t )  4 sin( 20t )
Discrete-Time Systems
• A discrete time system is a device or algorithm that operates on a
discrete time signals called input or excitation, according to some
well defined rule to produce another discrete time signal called
output or response.
• The method of producing the output is some times called
transformation.
y (n)  T x(n) or x(n)  y(n)
T
Example 2.21
The Accumulator
• A system whose output is the sum of the current input and all
previous inputs.
y ( n) 
n

k  
k 0
 x(k )   x(n  k )
• For instance if x(n)  [3 1 1 2 4] then the output of the

accumulator at n = 2 is: y(2) = x(2)+x(1)+x(0)+ x(-1)+x(-2)
• But what about values x(-3), x(-4), … ? Ofcourse we can consider
this system to be a device which is just turned on at n = -2. It
means that once it is on, the previous status or values of the input
has no impact on the system and the system is said to be initially
relaxed.
Realization of the accumulator
y ( n) 
y ( n) 
n

k  
k 0
 x(k )   x(n  k )
n 1
 x(k )  x(n)  y(n  1)  x(n)
k  
• It means that the current output depends on the current input and
the previous output.
• Note that y(n-1) summarizes the affect of all inputs till n-1.
• If we want to find the value of y(n) at n = n0 then,
y (n0 )  y (n0  1)  x(n0 )
y (n0  1)  y (n0 )  x(n0  1)
y (n0  2)  y (n0  1)  x(n0  2)



Realization of the accumulator contd.
• Now to evaluate y(n0-1), we have the formula:
y(n0  1) 
n0 1
 x( k )
k  
• Note that if the system is relaxed before n = n0 (even if it is not the
initial value), then y(n0-1) = 0.
• Example 2.2.2
Block diagram representation of discrete time
systems
Block diagram representation of discrete time
systems contd.
Example 2.2.3 contd.
Example 2.2.3 contd.
• Note that if we treat the system as input-output
(terminals only), then we are not concerned with the
actual implementation.
• If we consider the internal structure also, then we know
exactly how the input signal is handled.
• This system is relaxed at any sample time when all the
delays in the system are zero (memory is filled with
zero)
Properties and classification of discrete time systems
Static and dynamic systems:
• If the O/P depends on the current input but not the previous or future inputs.
(No memory). Generally it is indicated as:
y (n)  T x(n), n
e.g. y (n)  3  2 x(n), y(n)  x 3(n)  1
• Other than that, the system is called dynamic. (Memory is there)
• Finite memory of length N : in which the current O/P depends on the current
input and N previous inputs. e.g.
y ( n) 
N , N 0
 x(n  k )
This is a finite accumulato r of length N
k 0
n
y (n)   x(n  k ) This is a finite accumulato r of length n
k 0

y (n)   x(n  k ) This is an infinite accumulato r
k 0
Properties and classification of discrete time systems
Time variant and time-invariant:
How to check the time variance of a system:
1.
Find the expression of x(n) and y(n) for the relaxed system T.
2.
Find y(n-k) simply by replacing n by n-k in the y(n) expression.
3.
Find T[x(n-k)] = y(n,k) simply by replacing x(n) by x(n-k) in the y(n)
expresion.
4.
If y(n,k) = y(n-k) then the system is time invariant otherwise, even for a single
value of k, the system is time variant.
Example 2.2.4
Properties and classification of discrete time systems
contd.
Linear and non linear:
• A linear system is one which satisfies the superposition rule.
Linear and non linear contd.
• The scaling property of linear systems:
• The additive property of linear systems:
• General rule for linear systems:
Linear and non linear contd.
• Another General rule for linear systems: if the input to a
system T is 0, and if the output is not zero, then either the
system is non linear, or it is linear but non relaxed.
• Hence,
T is relaxed, superposition is satisfied
T is relaxed, superposition or I/P O/P not satisfied
T non relaxed, superposition is satisfied
T non relaxed, superposition not satisfied
• Example 2.2.5




linear
non linear
linear
non-linear
Properties and classification of discrete time systems
contd.
Causal versus non causal systems:
• But how can we have future terms like x(n+1). It is not realizable.
Certainly we can implement causal systems but how can we
implement non causal systems?
• Example 2.2.6
Properties and classification of discrete time systems
contd.
Stable and unstable systems:
• It is an important parameter when considering the practical
implementation of the discrete systems.
• Non stable systems causes unusual behavior and some times
results in incorrect outputs.
• Bounded means: x(n)  M x   produces  y(n)  M y  
• Example 2.2.7
Consider t he non linear system y (n)  y 2 (n  1)  x(n)
If the input is x(n)  C (n), then determine the stability of the system
Interconnection of Discrete time systems
Interconnection of Discrete time systems contd.
• Note: in cascaded system, if the systems are linear and time
invariant, then TC is also time invariant and T2T1  T1T2 otherwise
T2T1  T1T2
• We can use parallel and cascade interconnection of systems to
construct larger systems.
• We can break complex systems into smaller subsystems for the
purpose of analysis.
• This is used more frequently when designing and implementing
digital filters.
Analysis of Linear time invariant (LTI)
systems
• An important class of systems is LTI.
• The response of LTI systems to any input can be expressed in
terms of unit sample response.
• The main objective of the analysis of such systems is to find out a
relation (expression) between the input and the output discrete
signals.
• One of the techniques used for such analysis is by finding a
solution for the differential equations which characterize such
systems.
• The other technique is used in which the signal is decomposed into
basic signals, then the response of these basic signals are
evaluated. The total response of these signals is the response of the
actual signal.
Decomposition Method
• Let x(n) be the input signal to a linear system T.
• This signal is decomposed into different basic signals indicated as
ck xk (n) such that x(n)   ck xk (n)
k
where ck are the weighting coefficients
• If T is linear and relaxed then,
ck yk (n)  T ck xk (n)


then, y(n)  T x(n)  T  ck xk (n)
 k

And hence y(n)   ck yk (n)
k
Decomposition Method contd.
•
•
•
•
•
One of the basic signals being used is the impulse signals  (n)
Hence, our basic signal ck xk (n) becomes ck (n  k )
According to sifting property, x(n) (n  k )  x(k ) (n  k )
Hence, ck  x(k )
The input signal x(n) becomes:

 x(k ) (n  k )
k  
• Any signal is represented by
the weighted sum of impulse
signals. The weights are the
samples of the signal at k. see
example 2.3.1
Decomposition Method contd.
• Let h( n, k ) be the response of  (n  k ) for this linear system.
• Hence, the response of ck (n  k )  x(k ) (n  k ) is x( k ) h( n, k )
• Finally, the response of the input signal x(n) to a linear system T is
y(n) such that,
x ( n) 

T

 x(k ) (n  k )  y(n)   x(k )h(n, k )
k  
k  
• The response of the input signal x(n) to LTI system T is y(n) such
that,
x ( n) 

T

 x(k ) (n  k )  y(n)   x(k )h(n  k )
k  
k  
The convolution sum
• A relaxed LTI system is completely characterized by a single
function h(n) which is the response of the unit impulse  (n)
• For time variant linear system, we have to evaluate h( n, k ) for each
value of k
• The response of LTI system is called the convolution sum which
is written as:
y ( n)  x ( n)  h( n)
How does it work
• The convolution is a function of two variables, n (the actual sample
time) and k (used for delay and as a dummy variable when
evaluating the convolution for a particular value of n)
• If we want to evaluate the output y(n) at n  n0 , we perform the
following operations:
Example 2.3.2 : In LTI, h(n)  [1, 2,1,1] and x(n)  [1,2,3,1]


Another Way to visualize Convolution
x(k )
h( n  k )
Example 2.3.2 contd.
Properties of convolution
• It is irrelevant which of the two sequences is folded and shifted.
We can assume that h(n) is the input signal and x(n) is the
response. This is called the commutative law Hence:
x ( n)  h( n) 

 x ( k ) h( n  k ) 
k  
h( n)  x ( n) 

 h( k ) x ( n  k )
k  
• The only difference is in the intermediate multiplication but not the
final output.
• To find the output of a system from the given input and impulese
response, make a wise choice of which signal to fold and shift.
Example 2.3.3
Properties of convolution contd.
• Associative law (cascaded LTI systems):
x(n)  h1 (n) h2 (n)  x(n)  h1 (n)  h2 (n)
such that h(n)  h1 (n)  h2 (n) is the total response
• Example 2.3.4
Properties of convolution contd.
• Distributive law (parallel LTI systems):
x(n)  h1 (n)  h2 (n)  x(n)  h1 (n)  x(n)  h2 (n)
such that h(n)  h1 (n)  h2 (n) is the total response
• Generally
L
h(n)   h j (n)
j 1
Causal linear time-invariant systems
• It is the conventional LTI system but now the current output
depends on the current input and or the previous inputs. Let n  n0
be the sample at which we want to find the output. Hence,
• An LTI system is causal if and only if its impulse response is
zero for negative values of n. Solve 2.3.2 using matrix method
of causal LTI systems. (What changes are there?)
• In both previous equations, the input should be the current
and past ones and at the same time, h(n) exist only for n > 0.
Causal linear time-invariant systems contd.
• Another type of system is the causal LTI systems with causal
input sequences.
• Causal input sequence means x(n) = 0, for n < 0. Hence the output
for such systems become,
n
n
k 0
k 0
y ( n)   h( k ) x(n  k ) OR y (n)   x(k )h(n  k )
• The response of a causal system to a causal input sequence is a
causal sequence. i.e. y(n) = 0 for n < 0.
• Example 2.3.5: Determine the unit step response of the LTI
system with impulse response: h(n)  a nu(n)
a 1
Finite and infinite duration impulse response
• All LTI systems are divided into two groups: 1) With finite
impulse response (FIR), and 2) With infinite impulse response
(IIR).
• In FIR systems, the value of h(n) is zero outside some duration of
length M. As a special case, in causal FIR systems, the output
becomes:

M 1
k 0
n
k 0
n
y (n)   h(k ) x(n  k ) becomes y (n) 
y ( n) 
 x ( k ) h( n  k )
k  
becomes y (n) 
 h( k ) x ( n  k )
 x ( k ) h( n  k )
k  n  M 1
• Note that in causal-FIR, h(n) = 0, for 0  n and n  M
• The output at any time n is simply a weighted linear combination
of the input signal samples x(n), x(n  1), , x(n  M  1)
Finite and infinite duration impulse response
contd.
• This system acts as a window by considering the M most recent
inputs and neglecting the rest. Hence we say the causal FIR
systems have finite memory of length M.
• The IIR system has infinite duration and hence the expression
is same as the normal causal LTI systems.
• One last word:
A combination of input/output expression or input/impulse
response and ultimately input/impulse response/output
completely describe the behavior and the characteristics of the
system in terms of linearity, causality, time variance, etc…
.