Chapter 2

Discrete-Time Signals &

Systems

清大電機系林嘉文
cwlin@ee.nthu.edu.tw
03-5731152

Discrete-Time Signals
 Signals are represented as sequences of numbers, called
samples
 Sample value of a typical signal or sequence denoted as x
= {x[n]} with − ∞ ≤ n ≤ ∞
 x[n] is defined only for integer values of n and undefined for
non-integer values of n
 Representation of discrete-time signals:
 n  2, n  0
 Functional representation x  n  
 3, n  0
 Tabular representation
 Sequence representation
x(n) = {…,0.2, 2.2, 1.1, 0.2, -3.7, 2.9. …}

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Discrete-Time Signals
 Graphical representation

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Discrete-Time Signals
 Sampling a speech signal
x  n   xa (nT ),    n  

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2
Basic Sequences
1, n  0
 Unit sample sequence -   n   
0, n  0

1, n  0 n

 Unit step sequence - u  n   u  n    k 

0, n  0 k 

   n  k 
k 0

  n   u  n   u  n  1

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Basic Sequences
 n, n  0
 Unit ramp signal - ur  n   
0, n  0

 Real exponential signal -

x  n   A n  is a real value
0   <1  >1

-1    0   -1

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 Basic Sequences
 Complex exponential signal -
x  n   A n  A e j  e j n    e j
n
0 0

 A
n
 cos  n     j sin  n    
x  n   xR  n   jxI  n 
0 0

xR  n   A  cos 0 n   
n

xI  n   A  sin 0 n   
n

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Basic Sequences
 Complex exponential signal -
x  n   xR  n   jxI  n   x  n  x  n 

x  n  A n  r n

x  n     n    n

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Basic Sequences
 Sinusoidal signals with different frequencies

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Basic Sequences
 An arbitrary sequence can be represented in the time-
domain as a weighted sum of some basic sequence and
its delayed (advanced) versions

p n   p  k   n  k 
k 

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The Norm of a Discrete-Time Signal
 Size of a Signal - given by the norm of the signal
1
Lp-norm：   p p
x    x  n 
 n  
p

where p is a positive integer

 The value of p is typically 1 or 2 or ∞
L2-norm x 2 is the root-mean-squared (rms) value of {x[n]}
L1-norm x 1
is the mean absolute value of {x[n]}
L∞-norm x  is the peak absolute value of {x[n]} (why?)
x 
 x max

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Classification of Discrete-Time Signals

 Periodic signals and aperiodic signals
 A signal is periodic with period N (N > 0) if and only if
x  n  N   x  n  for all n
 The smallest value of N for which the above condition
holds is called the (fundamental) period

 A signal not satisfying the periodicity condition is

called nonperiodic or aperiodic

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Classification of Discrete-Time Signals

 Conjugate-symmetric sequence:
x  n   x*   n 

 If x[n] is real, then it is an even sequence

 for a conjugate-symmetric sequence {x[n]}, x[0]
must be a real number

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Classification of Discrete-Time Signals

 Conjugate-antisymmetric sequence:
x  n    x*   n 

 If x[n] is real, then it is an odd sequence

 for a conjugate anti-symmetric sequence {y[n]}, y[0]
must be an imaginary number

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Classification of Discrete-Time Signals
 Any complex sequence can be expressed as a sum of its
conjugate-symmetric and conjugate-antisymmetric parts:
x  n   xcs  n   xca  n 
where 
 xcs  n   2  x  n   x   n 
1 *


 x  n   1  x  n   x*   n  
 ca 2
 Any real sequence can be expressed as a sum of its even
part and its odd part:
x  n   xev  n   xod  n 
where 
 xev  n   2  x  n   x  n 
1


 x  n   1  x  n   x   n 
 od 2
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Classification of Discrete-Time Signals

 Periodic signals and aperiodic signals
 A signal is periodic with period N (N > 0) if and only if
x  n  N   x  n  for all n
 The smallest value of N for which the above condition
holds is called the (fundamental) period

 A signal not satisfying the periodicity condition is

called nonperiodic or aperiodic

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Classification of Discrete-Time Signals
 Energy signals and power signals
 The total energy of a signal x(n) is defined by


 x  n
2
E
n 

 An infinite length sequence with finite sample values

may or may not be an energy signal (with finite energy)
 The average power of a discrete-time signal x[n] is
defined by 1 N

 x  n
2
P  lim
N  2 N  1 n  N
 Define the signal energy of x(n) over the finite interval
− N ≤ n ≤ N as N

 x n
2
EN 
n  N

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Classification of Discrete-Time Signals

 Energy signals and power signals
 The signal energy can then be expressed as
E  lim EN
N 

 The average power of x(n) becomes

1
P  lim EN
N  2 N  1

 If E is finite, P = 0. On the other hand, if E is infinite,

the average power P may be either finite or infinite
 If P is finite (and nonzero), the signal is called a power
signal

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Classification of Discrete-Time Signals
 Energy signals and power signals
 Example – Determine the power and energy of the
unit step sequence
The average power of the unit step signal is
1 N
N 1 1
P  lim
N  2 N  1

n 0
1  lim
N  2 N  1

2
It’s a power signal with infinite energy
 Example - Consider the causal sequence defined by
3  1 , n  0
n

x  n  
 0, n0
Note: x(n) has infinite energy, its average power is
1  N 
P  lim 91  4.5
N  2 N  1 
 n 0 
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Classification of Discrete-Time Signals

 An infinite energy signal with finite average power is
called a power signal
 Example - A periodic sequence which has a finite
average power but infinite energy
 A finite energy signal with zero average power is called
an energy signal
 Example - A finite-length sequence which has finite
energy but zero average power

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 Classification of Discrete-Time Signals

 A sequence x[n] is said to be bounded if

x  n   Bx  
 Example - The sequence x[n] = cos0.3πn is a bounded
sequence as
x  n   cos(0.3 n)  1

 A sequence x[n] is said to be absolutely summable if



 x  n  
n 

 Example - The following sequence is absolutely

summable 0.3n , n  0
y[n]  
 0, n0

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Classification of Discrete-Time Signals

 A sequence x[n] is said to be square summable if


 x  n
2

n 

 Example - The sequence

sin(0.4 )
h[n] 
n
is square-summable but not absolutely summable

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Manipulation of Discrete-Time Signals (1/5)
 Transformation of independent variable (time)
 Time shifting: A signal x[n] may be shifted in time by
replacing the independent variable n by n – k

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Manipulation of Discrete-Time Signals (2/5)

 Transformation of independent variable (time)
 Folding/Reflection: A signal x[n] may be folded in
time by replacing the independent variable n by –n

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Manipulation of Discrete-Time Signals (3/5)
 The operations of folding and time delaying (or
advancing) a signal are NOT commutative
 Denote the time-delay operation by TD and the folding
operation by FD
TDk  x  n   x  n  k  , k 0

FD  x  n   x  n 

Now
 
TDk FD  x  n   TDk  x   n   x   n  k 

whereas

   
FD TDk  x  n   FD  x  n  k   x   n  k   TDk FD  x  n 

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Manipulation of Discrete-Time Signals (4/5)

 Transformation of independent variable (time)
 Time Scaling or down-sampling: A signal x[n] may
be scaled in time by replacing n by n

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Manipulation of Discrete-Time Signals (5/5)
 Transformation of independent variable (time)
 Addition, multiplication, and scaling of sequences:
Amplitude modifications include addition, multiplication,
and scaling of discrete-time
 Amplitude scaling of a signal by a constant :
y  n   Ax  n  ,   n  
 Sum of two signals:
y  n   x1  n  +x2  n  ,   n  
 Product of two signals:
y  n   x1  n  x2  n  ,   n  

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Discrete-Time Systems
 Discrete-time system: A device or an algorithm that
performs some prescribed operation on a discrete-time
signal (input or excitation) to produce another discrete-
time signal (output or response)

 We say that the input signal x[n] is “transformed” by the

system into a signal y[n] as expressed below

y  n   T  x  n 

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Input-Output Description of Systems
 The input-output description or a discrete-time system
consists of a mathematical expression or a rule, which
explicitly defines the relation between the input and
output signals
x  n  
T
 y  n
 Example: Determine the response of the following
systems to the input signal
 n , 3  n  3
x  n  
 0, otherwise
(b) y  n    x  n  1  x  n   x  n  1
1
(a) y  n   x  n  1
3
n

(c) y  n   median  x  n  1 , x  n  , x  n  1 (d) y  n    x k 

k 

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Linear Systems: Accumulator

n

 Accumulator - y[n]   x[l ]

l 
n 1
  x[l ]  x[n]  y[n  1]  x[n]
l 
 The output y[n] is the sum of the input sample x[n] and
the previous output y[n −1]
 The system cumulatively adds, i.e., it accumulates all
input sample values
 Input-output relation can also be written in the form
1 n n
y[n]   x[l ]   x[l ]  y[1]   x[l ],
l  l 0 l 0
n0

 The second form is used for a causal input sequence, in

which case y[−1] is called the initial condition
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Linear Systems: Moving Average
M2
1
y n   x n  k 
M 1  M 2  1 k  M1

 x  n  M1   x  n  M1  1  ...  x  n  x  n  1  ...  x  n  M 2 
1

M1  M 2  1

 An application: Consider x[n] = s[n] + d[n] where s[n] =

2[n(0.9)n] is the signal corrupted by a random noise d[n]

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Nonlinear Systems: Median Filter (1/3)

 The median of a set of (2K+1) numbers is the number
such that K numbers from the set have values greater than
this number and the other K numbers have values smaller
 Median can be determined by rank-ordering the numbers
in the set by their values and choosing the number at the
middle
 Example: Consider the set of numbers
{2, −3, 10, 5, −1}
 Rank-order set is given by
{−3 , −1, 2 , 5, 10}
 median{2, −3, 10, 5, −1} = 2

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Nonlinear Systems: : Median Filter (2/3)
 Median Filtering Example

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Nonlinear Systems: Median Filter (3/3)

 Median Filtering Example

Original Image Noisy Image Filtered Image

(pepper-and-salt noise)

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Block Diagram Representation of
Discrete-Time Systems
 Adder

 Constant multiplier

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Block Diagram Representation of

Discrete-Time Systems
 Signal multiplier/Modulator

 Unit delay element

 Unit advance element

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Block Diagram Representation of
Discrete-Time Systems
1 1 1
 Example: y  n  y  n  1  x  n   x  n  1
4 2 2

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Static vs. Dynamic Systems

 A discrete-time system is called static or memoryless if
its output at any time instant depends at most on the
input sample at the same time y  n   T  x  n , n
y  n   ax  n 
y  n   nx  n   bx3  n 
 If a discrete-time system is not static, it is said to be
dynamic or to have memory

y  n   x  n   3 x  n  1 (finite memory)

y  n   x  n  k  (infinite memory)
k 0

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Time (Shift) Invariance
 Time-invariant vs. time-variant systems
 A system is called time-invariant if its input-output
characteristics do not change with time y  n   T  x  n
 Definition: A relaxed system T is time-invariant or
shift-invariant if and only if
x(n) 
T
 y ( n)
Implies that x(n  k ) 
T
 y (n  k )

For every input signal x(n) and every time shift k.

 In general, we can write the output of a time-invariant
system as
y (n, k )  T  x(n  k ) 

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Time (Shift) Invariance

 Examples y  n   T  x  n   x  n   x  n  1

y  n, k   x  n  k   x  n  k  1
y  n  k   x  n  k   x  n  k  1
y  n, k   y  n  k 
time invariant

y  n   T  x  n   nx  n 

y  n, k   nx  n  k 
y n  k   n  k  x n  k 

time variant
y  n, k   y  n  k 

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Time (Shift) Invariance
 Examples y  n   T  x  n   x  n 

y  n, k   x  n  k 
y  n  k   x  n  k 

time variant y  n, k   y  n  k 

y  n   T  x  n   x  n  cos 0 n
y  n, k   x  n  k  cos 0 n
y  n  k   x  n  k  cos 0  n  k 
y  n, k   y  n  k 
time variant

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Linearity (1/3)
 A linear system is one that satisfies the superposition
principle
 Definition: A system T is linear if and only if

T a1 x1  n  a2 x2  n   a1T  x1  n   a2T  x2  n 

for any arbitrary input sequences x1[n] and x2[n], and any
arbitrary constants a1 and a2.
 Multiplicative/scaling property: Suppose that a2 = 0
T a1 x1  n   a1T  x1  n   a1 y1  n 
 Additivity property: Suppose that a1 = a2 = 1
T  x1  n   x2  n   T  x1  n  T  x2  n  y1  n  y2  n

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Linearity (2/3)
 Graphical representation of the superposition principle

T is linear if and only if y[n] = y’[n]

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Linearity (3/3)
 Linear vs. non-linear systems
 The linear condition can be extended arbitrarily to any
weighted linear combination of signals
M 1 M 1
x  n    ak xk  n  
T
 y  n    ak yk  n 
k 1 k 1
where
yk  n   T  xk  n  , k  1, 2, , M  1

 If a system produces a nonzero output with a zero

input, it may be either non-relaxed or nonlinear
 Examples: (a) y[n] = nx[n], (b) y[n] = x[n2], (c) y[n] =
x2[n], (d) y[n] = Ax[n] + B, (e) y[n] = ex[n]

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Causality
 Causal vs. non-causal systems
 Definition: A system is said to be causal if the output
of the system at any time n depends only on present
and past inputs, but does not depend on future inputs
y  n   T  x  n  , x  n  1 , x  n  2 ,

where T{·} is some arbitrary function.

 Noncausal vs. anticausal
 If a system produces a nonzero output with a zero
input, it may be either non-relaxed or nonlinear
 Examples: (a) y[n] = x[n]  x[n  1], (b) y[n] = x[n] +
3x[n+4], (c) y[n] = x[n2], (d) y[n] = x[2n], (e) y[n] = x[n]

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Stability
 Bounded-Input, Bounded Output (BIBO) stability
If y[n] is the response to an input x[n] and if
x[n]  Bx for all values of n
then
y[n]  By for all values of n
 Example – the M-point moving average filter is BIBO
stable 1 M 1
y[n] 
M
 x[n  k ]
k 0

 With a bounded input x[n]  Bx

M 1 M 1
1 1
y[n] 
M
 x[n  k ]  M  x[n  k ]
k 0 k 0

1
  MBx   Bx
M

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23
Passive & Lossless Systems
 A discrete-time system is defined to be passive if, for
every finite-energy input x[n], the output y[n] has, at
most, the same energy
 

 y[n]   x[n]
2 2

n   n  
 For a lossless system, the above inequality is satisfied
with an equal sign for every input
 Example - Consider the discrete-time system defined by
y[n] =α x[n − N] with N a positive integer
 Its output energy is given by
 

 y[ n]   
2 2 2
x[ n]
n  n 

passive system if ǀαǀ <1, and lossless if ǀαǀ =1

2011/3/2 Digital Signal Processing 47

Interconnection of Discrete-Time Systems

 Cascade interconnection

y1  n   T1  x  n 
y  n   T2  y1  n   T2 T1  x  n  
 Systems T1 and T2 can be combined or consolidated
into a single overall system
y  n   Tc  x  n  where Tc  T2T1
 In general TT1 2  T2T1 . However, if systems T1 and T2
are LTI, then (a) is time invariant and (b) TT
1 2  T2T1

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24
Interconnection of Discrete-Time Systems
 Parallel interconnection

y3  n   y1  n   y2  n 
 T1  x  n   T2  x  n 
  T1  T2   x  n 
 Tp  x  n 
 We can use parallel and cascade interconnection of
systems to construct larger, more complex systems
2011/3/2 Digital Signal Processing 49

Techniques for the Analysis of Linear

Systems
 Two basic methods for analyzing the behavior of a linear
system:
 The first is based on the direct solution of the input-
output equation
N M
y  n    ak y  n  k    bk x  n  k 
k 1 k 0

 The second method is to decompose or resolve the

input signal into a sum of elementary signals. Then,
using the linearity of the system, the response of the
system to the elementary signals are sum to obtain
the total response

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25
Techniques for the Analysis of Linear
Systems
 Suppose the input signal is resolved into a weighted
sum of elementary signals
x  n    ck xk  n 
k

 The response yk[n] of the system to the component

xk[n] is
yk  n   T  xk  n 

 If the system is linear, we have

 
y  n   T  x  n  = T  ck xk  n 
 k 
  ck T  xk  n    ck yk  n 
k k
Why & how to do the signal decomposition?
2011/3/2 Digital Signal Processing 51

Resolution of a Discrete-Time Signal into

Impulses
 Select the elementary signals xk[n] to be
xk  n     n  k 
where k represents the delay of the unit sample sequence
 Multiply the two sequences x[n] and [nk]?
x  n  n  k   x  k   n  k 

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26
Resolution of a Discrete-Time Signal into
Impulses
 Consequently

x  n   x  k   n  k 
k 

 Example - Consider a finite-duration sequence given as

x  n   2, 4, 0,3
The sequence can be resolved as

x  n   2  n  1  4  n   3  n  2

2011/3/2 Digital Signal Processing 53

Resolution of a Discrete-Time Signal into

Impulses
 The response of a relaxed linear system to the unit sample
sequence input:
y  n, k   h  n, k   T   n  k 

 If the impulse at the input is scaled by as

ck h  n, k   x  k  h  n, k 
 If the input is expressed as 
x  n   x  k   n  k 
k 

The output becomes

  
y  n   T  x  n   T   x  k    n  k 
 k  
 
  x  k  T   n  k    x  k  h  n, k 
k  k 

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27
Response of LTI Systems to Arbitrary
Inputs
 If the system is time invariant, and denote the response of
the LTI system to the unit sample sequence as
h[n]  T  [n  k ]
 The response of the system to   n  k  is
h  n  k   T   n  k 
 Consequently 
y  n   x k  h n  k 
k 

 The relaxed LTI system is completely characterized by a

single function h[n], the impulse response.
 Convolution is commutative
 
y  n   x k  h n  k    h k  x n  k 
k  k 

2011/3/2 Digital Signal Processing 55

Computing the Convolution Sum

 The output of an LTI system at n = n0 is given by

y  n0    x k  h n 0  k
k 
 To compute y[n0]
 Folding. Fold h[k] about k = 0 to obtain h[k]
 Shifting. Shift h[k] by n0 to the right (left) if is positive
(negative), to obtain h[n0k]
 Multiplication. Multiply x[k] by h[n0k] to obtain the
product sequence
vn0  k   x  k  h  n0  k 
 Summation. Sum all the values of vn  k  to obtain y[n0] 0

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28
Computing the Convolution Sum
x  n   1, 2,3,1 h  n   1, 2,1, 1

y  n    , 0, 0,1, 4,8,8,3, 2, 1, 0, 0,

2011/3/2 Digital Signal Processing 57

Computing the Convolution Sum

  
y[n]  x[n]* h[n]    x[k ] [n  k ]  * h[n]   x2 [n]  x0 [n]  x3 [n] * h[n]
 k  
  x[2] [n  2]  x[0] [n]  x[3] [n  3] * h[n]
 x[2]( [n  2]* h[n])  x[0]( [n]* h[n])  x[3]( [n  3]* h[n])
 x[2]h[n  2]  x[0]h[n]  x[3]h[n  3]  y2 [n]  y0 [n]  y3[n]

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29
Computing the Convolution Sum

2011/3/2 Digital Signal Processing 59

Tabular Method of Convolution Sum

Computation
N2
y[n]  g[n] * h[n]   g[n  k ]h[k ]
k  N1
n N2 n N 2
  g[k ]h[n  k ] 
k  n  N1
 g[k ]h[(k  n)]
k  n  N1

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30
 Computing the Convolution Sum
 Example:

h  n  a nu  n , a  1
x  n  u  n
y  0  1
y 1  1  a
y  2  1  a  a 2

y  n  1  a  a 2    a n
1  a n 1

1 a
1
y     lim y  n  
n  1 a
2011/3/2 Digital Signal Processing 61

Computing the Convolution Sum

h[n]  u[n]  u[n  N ]
x[n]  a nu[n]

y[n]  x[n] * h[n]   h[k ]x[n  k ]
k 

 a nk
u[n  k ]u[k ]  u[k  N ]  a n k

k  k n,k 0,k  N


0, if n  0

n an (1  a ( n1) ) n k
  a nk   a if 0  n  N 1
k 0 1  a 1 k 0
N 1 nk an (1  a  N )  1 a N 
 a  1
 an N 1 , if n  N
k 0 1 a  1 a 
M
r N  r M 1
Note:  r k 
k N 1 r
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31
Properties of Convolution (1/2)
 Commutative Property

y  n  x  n  h  n   x k  h n  k 
k 

 h  n  x  n   h k  x n  k 
k 

 Identity and Shifting Properties

y  n  x  n    n  x  n

x  n    n  k   y n  k   x  n  k 

2011/3/2 Digital Signal Processing 63

Properties of Convolution (2/2)

 Associative Property

 x  n   h  n   h  n    x  n   h  n   h  n   x  n    h  n   h  n 
1 2 2 1 1 2

 Distributive Property
x  n    h1  n   h2  n   x  n   h1  n   x  n   h2  n 

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32
 Causality of LTI Systems (1/2)
 The output of an LTI system at n = n0 is given by

y  n0    x k  h n 0  k
k 

 Divide

the sum into
1
two sets of terms:
y  n0    h  k  x  n0  k    h  k k  x n 0  k
k 0 k 

   
  h  0 x  n0   h 1 x  n0  1     h  1 x  n0  1  h  2 x  n0  2  
      
 depend on present and past inputs   depend on future inputs 
 For a causal system, h[n] = 0 for n < 0
 Since h[n] is the response of the relaxed LTI system to a
unit impulse sequence at n = 0, an LTI system is causal
if and only if its impulse response is zero for negative
values of n
2011/3/2 Digital Signal Processing 65

Causality of LTI Systems (2/2)

 The output of an causal LTI system becomes
 n
y  n   h  k  x  n  k    x k  h n  k 
k 0 k 
 A sequence x[n] is called a causal sequence if x[n] = 0
for n < 0; otherwise, it’s a noncausal sequence
 If the input to a causal LTI system is a causal sequnce,
the input-output equation reduces to
n n
y  n   h  k  x  n  k    x k  h  n  k 
k 0 k 0

 Example: Determine the unit step response of the LTI

system with impulse response
h  n  a nu  n  , a 1
n
1  a n 1
y  n   a k 
k 0 1 a
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Stability of LTI Systems (1/3)
 BIBO Stability Condition - A discrete-time system is
BIBO stable if and only if the output sequence {y[n]}
remains bounded for all bounded input sequence {x[n]}
 An LTI discrete-time system is BIBO stable if and only if
its impulse response sequence {h[n]} is absolutely
summable, i.e. 
Bh   h k   
k 

 Proof: Assume h[n] is a real sequence

Sufficient condition: Since the input sequence x[n] is
bounded we have x(n)  Bx  
therefore
  
y[n]  
k 
h[k ]x[ n  k ]  
k 
h[k ] x[ n  k ]  Bx 
k 
h[ k ]  Bx Bh  

2011/3/2 Digital Signal Processing 67

Stability of LTI Systems (2/3)

 Thus, Bh < ∞ implies ǀy[n]ǀ ≤ BxBh < ∞, indicating that y[n] is
also bounded
 To prove the necessary condition, assume y[n] is bounded,
i.e., ǀy[n]ǀ ≤ By
 Consider the bounded input given by
 h*   n 
 , h  n  0
x  n   h  n

 0, h n  0
 For this input, y[n] at n = 0 is
h k 
2
 
y  0   x  k  h  k    h  k   Bh
k  k 

 Therefore, if Bh = ∞, then {y[n]} is not a bounded sequence

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Stability of LTI Systems (3/3)
 Example - Consider a causal LTI discrete-time system
with an impulse response
h  n  a n u  n
 For this system
 
1
Bh   ak u  k    a  , if a  1
k

k  k 0 1 a

 Therefore Bh < ∞ if |a| < 1 , for which the system is BIBO

stable
 If |a| = 1, the system is not BIBO stable

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35