Chapter 2
Discrete-Time
Signals and Systems
§ 2.1 Discrete-Time Signals:
Time-Domain Representation
Signals represented as sequences of
numbers,called samples
Sample value of a typical signal or
sequence denoted as x[n] with n being an
integer in the range -n
x[n] defined only for integer values of n
and undefined for noninteger values of n
Discrete-time signal represented by {x[n]}
§ 2.1 Discrete-Time Signals:
Time-Domain Representation
Discrete-time signal may also be written
as a sequence of numbers inside braces:
{x[n]}={…,-0.2,2.2,1.1,0.2,-3.7,2.9,…}
The arrow is placed under the sample at
time index n = 0
In the above,x[-1]= -0.2,x[0]=2.2,
x[1]=1.1,etc,
§ 2.1 Discrete-Time Signals:
Time-Domain Representation
Graphical representation of a
discrete-time signal with real-
valued samples is as shown below:
§ 2.1 Discrete-Time Signals:
Time-Domain Representation
In some applications,a discrete-time
sequence {x[n]} may be generated by
periodically sampling a continuous-time
signal xa(t) at uniform intervals of time
§ 2.1 Discrete-Time Signals:
Time-Domain Representation
Here,n-th sample is given by
x[n]=xa(t) |t=nT=xa(nT),n=…,-2,-1,0,1,…
The spacing T between two consecutive
samples is called the sampling interval or
sampling period
Reciprocal of sampling interval T,denoted as
FT,is called the sampling frequency:
FT=1/T
§ 2.1 Discrete-Time Signals:
Time-Domain Representation
Unit of sampling frequency is cycles per
second,or hertz (Hz),if T is in seconds
Whether or not the sequence {x[n]} has been
obtained by sampling,the quantity x[n] is
called the n-th sample of the sequence
{x[n]} is a real sequence,if the n-th sample x[n]
is real for all values of n
Otherwise,{x[n]} is a complex sequence
§ 2.1 Discrete-Time Signals:
Time-Domain Representation
A complex sequence {x[n]} can be
written as {x[n]}={xre[n]}+j{xim[n]}
where xre and xim are the real and
imaginary parts of x[n]
The complex conjugate sequence of {x[n]}
is given by {x*[n]}={xre[n]} - j{xim [n]}
Often the braces are ignored to denote a
sequence if there is no ambiguity
§ 2.1 Discrete-Time Signals:
Time-Domain Representation
Example - {x[n]}={cos0.25n} is a
real sequence {y[n]}={ej0.3n} is a
complex sequence
We can write
{y[n]}={cos0.3n + jsin0.3n}
= {cos0.3n} + j{sin0.3n}
where {yre[n]}={cos0.3n}
{yim[n]}={sin0.3n}
§ 2.1 Discrete-Time Signals:
Time-Domain Representation
Two types of discrete-time signals,
- Sampled-data signals in which samples
are continuous-valued
- Digital signals in which samples are
discrete-valued
Signals in a practical digital signal
processing system are digital signals
obtained by quantizing the sample
values either by rounding or truncation
§ 2.1 Discrete-Time Signals:
Time-Domain Representation
Example -
Am
pli
tud
e
Digital signal
Am
pli
tud
e
Boxedcar signal
Time,t Time,t
§ 2.1 Discrete-Time Signals:
Time-Domain Representation
A discrete-time signal may be a finite-
length or an infinite-length sequence
Finite-length (also called finite-duration
or finite-extent) sequence is defined only
for a finite time interval:N1?n?N2
where -?< N1 and N2 <? with N1? N2
Length or duration of the above finite-
length sequence is N= N2 - N1+ 1
§ 2.1 Discrete-Time Signals:
Time-Domain Representation
Example - x[n]=n2,-3?n? 4 is a
finite-length sequence of length
4 – (-3) + 1 = 8
y[n]=cos0.4n is an infinite-length
sequence
§ 2.1 Discrete-Time Signals:
Time-Domain Representation
A right-sided sequence x[n] has zero-
valued samples for n < N1
A right-sided sequence
If N1? 0,a right-sided sequence is called
a causal sequence
§ 2.2 Operations on Sequences
A single-input,single-output discrete-
time system operates on a sequence,
called the input sequence,according
some prescribed rules and develops
another sequence,called the output
sequence,with more desirable
properties
x[n] y[n]
Input sequence Output sequence
Discrete-time
system
§ 2.2 Operations on Sequences
For example,the input may be a signal
corrupted with additive noise
Discrete-time system is designed to
generate an output by removing the
noise component from the input
In most cases,the operation defining a
particular discrete-time system is
composed of some basic operations
§ 2.2.1 Basic Operations
Product (modulation) operation:
Modulator
x[n] y[n]
w[n]
y[n]=x[n].w[n]
An application is in forming a finite-length
sequence from an infinite-length sequence by
multiplying the latter with a finite-length
sequence called an window sequence
The process is called windowing
§ 2.2.1 Basic Operations
Addition operation:
x[n] y[n]
w[n]
y[n]=x[n]+w[n]–Adder
A
x[n] y[n] y[n]=A.x[n]–Multiplier
Multiplication operation
§ 2.2.1 Basic Operations
Time-shifting operation,where N is an integer
If N > 0,it is delaying operation
1?z y[n]x[n]
–Unit delay
y[n]=x[n-1]
y[n]x[n] z
-Unit advance
y[n]=x[n+1]
If N < 0,it is an advance operation
§ 2.2.1 Basic Operations
Time-reversal (folding) operation:
y[n]=x[-n]
Branching operation,Used to provide
multiple copies of a sequence
x[n] x[n]
x[n]
§ 2.2.1 Basic Operations
Example - Consider the two following
sequences of length 5 defined for 0?n?4,
{a[n]}={3 4 6 –9 0}
{b[n]}={2 –1 4 5 –3}
New sequences generated from the
above two sequences by applying the
basic operations are as follows:
§ 2.2.1 Basic operations
{c[n]}={a[n].b[n]}={6 –4 24 –45 0}
{d[n]}={a[n]+b[n]}={5 3 10 –4 -3}
{e[n]}=(3/2){a[n]}={4.5 6 9 –13.5 0}
As pointed out by the above example,
operations on two or more sequences
can be carried out if all sequences
involved are of same length and defined
for the same range of the time index n
§ 2.2.1 Basic operations
However if the sequences are not of
same length,in some situations,this
problem can be circumvented by
appending zero-valued samples to the
sequence(s) of smaller lengths to make
all sequences have the same range of the
time index
Example - Consider the sequence of
length 3 defined for 0?n? 2,{f[n]}={-2,1,-3}
§ 2.2.1 Basic Operations
We cannot add the length-3 sequence
to the length-5 sequence {a[n]} defined
earlier
We therefore first append {f[n[} with 2
zero-valued samples resulting in a
length-5 sequence {fe[n[}={-2 1 –3 0 0}
Then
{g[n]}={a[n]}+{f[n]}={1 5 3 –9 0}
§ 2.2.1 Basic Operations
Example -
y[n]=?1x[n]+?2x[n-1]+?3[n-2]+?4x[n-3]
§ 2.2.2 Sampling Rate Alteration
Employed to generate a new sequence
y[n] with a sampling rate F’T higher or
lower than that of the sampling rate FT
of a given sequence x[n]
Sampling rate alteration ratio is
R= F’T / FT
If R > 1,the process called interpolation
If R < 1,the process called decimation
§ 2.2.3 Classification of
Sequences based on periodicity
Example -
A sequence satisfying the periodicity
condition is called an periodic sequence
§ 2.2.4 Classification of
Sequences Energy and Power
Signals
Total energy of a sequence x[n] is
defined by
n
nx 2x ][?
An infinite length sequence with finite
sample values may or may not have finite
energy
A finite length sequence with finite
sample values has finite energy
§ 2.2.4 Classification of
Sequences Energy and Power
Signals
The average power of an aperiodic
sequence is defined by
K
KnKK
nxP 212 1x ][lim
K
KnKx
nx 2,][?
Define the energy of a sequence x[n] over
a finite interval -K? n? K as
§ 2.2.4 Classification of
Sequences Energy and Power
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
§ 2.2.4 Classification of
Sequences Energy and Power
Signals
Example - Consider the causal sequence
defined by
00
013
n
nnx n
,
,)(][
Note,x[n] has infinite energy
Its average power is given by
5.412 )1(9lim1912 1lim
0
K
K
KP K
K
nK
x
§ 2.2.4 Classification of Sequences
bounded,absolutely summable and
squaresummable
A sequence x[n] is said to be bounded if
xBnx ][
Example - The sequence x[n]=cos(0.3?n)
is a bounded sequence as
13.0c o s][ nnx
§ 2.2.4 Classification of Sequences
bounded,absolutely summable and
squaresummable
A sequence x[n] is said to be absolutely
summable if
n
nx ][
Example - The sequence
00
030
n
nny n
,
,.][
is an absolutely summable sequence as
4 2 8 5 71301 130
0
...
n
n
§ 2.2.4 Classification of Sequences
bounded,absolutely summable and
squaresummable
A sequence x[n] is said to be square-
summable if
n
nx 2][
Example - The sequence
n
nnh
4.0s i n][
is square-summable but not absolutely
summable
§ 2.3 Basic Sequences
Unit sample sequence -
0,0
0,1][
n
nn?
0,0
0,1][
n
nn?
Unit step sequence -
§ 2.3 Basic Sequences
Real sinusoidal sequence -
x[n]=Acos(?0n+?)
where A is the amplitude,?0 is the angular
frequency,and? is the phase of x[n]
Example -
§ 2.3 Basic Sequences
Exponential sequence -
,][ nAnx n
,)( oo je, jeAA
],[][][ )( nxjnxeeAnx imrenjj oo
),c o s(][ neAnx onre o
)sin (][ neAnx onim o
where
then we can express
If we write
where A and? are real or complex numbers
§ 2.3 Basic Sequences
xre[n] and xim[n] of a complex exponential
sequence are real sinusoidal sequences with
constant (?0=0),growing (?0>0),and decaying
(?0<0) amplitudes for n > 0
njnx )e x p (][ 6121
Real part Imaginary part
§ 2.3 Basic Sequences
Real exponential sequence -
x[n]=A?n,-?< n <?
where A and? are real numbers
=1.2?=0.9
§ 2.3 Basic Sequences
Sinusoidal sequence Acos(?0n +?) and
complex exponential sequence Bexp(j?0n)
are periodic sequences of period N if?0N=2?r
where N and r are positive integers
Smallest value of N satisfying?0N=2?r
is the fundamental period of the sequence
To verify the above fact,consider
x1[n]= Acos(?0n +?)
x2[n]= Acos(?0 ( n+N) +?)
§ 2.3 Basic Sequences
Now
x2[n]= cos(?0 ( n+N) +?)
= cos(?0n+?)cos?0N - sin(?0n+?)sin?0N
which will be equal to cos(?0n+?)=x1[n]
only if sin?0N= 0 and cos?0N = 1
These two conditions are met if and only
if?0N= 2?r or 2?/?0 = N/r
§ 2.3 Basic Sequences
If 2?/?0 is a noninteger rational number,
then the period will be a multiple of
2?/?0
Otherwise,the sequence is aperiodic
Example - x[n]=sin(?3n+?) is an
aperiodic sequence
§ 2.3 Basic Sequences
Here?0 = 0.1?
Hence N= 2?r/?0 = 20 for r = 1
0 = 0.1?
§ 2.3 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
]2[]1[5.1]2[5.0][ nnnnx ]6[75.0]4[ nn
§ 2.4 The Sampling Process
Often,a discrete-time sequence x[n] is
developed by uniformly sampling a
continuous-time signal xa(t) as indicated below
The relation between the two signals is
x[n]=xa(t)|t=nT=xa(nT),n=…,-2,-1,0,1,2,…
§ 2.4 The Sampling Process
Time variable t of xa(t) is related to the time
variable n of x[n] only at discrete-time instants
tn given by
tn= nT = n/FT = 2?n/?T
with FT=1/T denoting the sampling frequency
and?T= 2?FT denoting the sampling angular
frequency
§ 2.4 The Sampling Process
Consider the continuous-time signal
)c o s ()2c o s ()( tAtfAtx oo
)2c o s()c o s(][ nAnTAnx
T
oo
)c o s ( nA o
ToToo /2
is the normalized digital angular frequency of
x[n]
where
The corresponding discrete-time signal is
§ 2.4 The Sampling Process
If the unit of sampling period T is in
seconds
The unit of normalized digital angular
frequency?0 is radians/sample
The unit of normalized analog angular
frequency?0 is radians/second
The unit of analog frequency f0 is hertz
(Hz)
§ 2.4 The Sampling Process
The three continuous-time signals
)6c o s ()(1 ttg
)14c o s ()(2 ttg
)26c o s ()(3 ttg
)6.0c o s (][1 nng
)4.1c o s (][2 nng
)6.2c o s (][3 nng
of frequencies 3 Hz,7 Hz,and 13 Hz,are
sampled at a sampling rate of 10 Hz,i.e,with T =
0.1 sec,generating the three sequences
§ 2.4 The Sampling Process
Plots of these sequences (shown with circles)
and their parent time functions are shown
below:
Note that each sequence has exactly the same
sample value for any given n
0 0.2 0.4 0.6 0.8 1
-1
- 0.5
0
0.5
1
t i m e
A
m
pl
i
t
ude
§ 2.4 The Sampling Process
This fact can also be verified by observing that
)6.0c o s ()6.02(c o s)4.1c o s (][2 nnnng
)6.0c o s ()6.02(c o s)6.2c o s (][3 nnnng
As a result,all three sequences are identical
and it is difficult to associate a unique
continuous-time function with each of these
sequences
§ 2.4 The Sampling Process
The above phenomenon of a continuous-
time signal of higher frequency
acquiring the identity of a sinusoidal
sequence of lower frequency after
sampling is called aliasing
§ 2.4 The Sampling Process
Recall
0=20/?T
Thus if?T>2?0,then the corresponding
normalized digital angular frequency?0 of the
discrete-time signal obtained by sampling the
parent continuous-time sinusoidal signal will
be in the range -?<?<?
Conclusion,No aliasing
§ 2.4 The Sampling Process
On the other hand,if?T < 2?0,the
normalized digital angular frequency will
foldover into a lower digital frequency
0 = ( 20 /?T)2? in the range -?<?<?
because of aliasing
Hence,to prevent aliasing,the sampling
frequency?T should be greater than 2 times
the frequency?0 of the sinusoidal signal being
sampled
§ 2.4 The Sampling Process
Generalization,Consider an arbitrary
continuous-time signal xa(t) composed of a
weighted sum of a number of sinusoidal
signals
xa(t) can be represented uniquely by its
sampled version {x[n]} if the sampling
frequency?T is chosen to be greater than 2
times the highest frequency contained in xa(t)
The condition to be satisfied by the sampling
frequency to prevent aliasing is called the
sampling theorem
§ 2.5 Discrete-Time Systems
A discrete-time system processes a given
input sequence x[n] to generates an
output sequence y[n] with more
desirable properties
In most applications,the discrete-time
system is a single-input,single-output
system:
x[n] y[n]
Input sequence Output sequence
Discrete-Time
System
§ 2.5 Discrete-Time Systems
2-input,1-output discrete-time systems -
Modulator,adder
1-input,1-output discrete-time systems -
Multiplier,unit delay,unit advance
§ 2.5 Discrete-Time Systems
Accumulator,
n xny
][][
][]1[][][1 nxnynxxn
The output y[n] at time instant n is the sum of
the input sample x[n] at time instant n and the
previous output y[n-1] at time instant n-1
which is the sum of all previous input sample
values from -? to n-1
The system accumulatively adds,i.e.,it
accumulates all input sample values
§ 2.5 Discrete-Time Systems
Accumulator - Input-output relation can
also be written in the form
n xxny
0
1 ][][][
,][]1[
0
n xy
0?n
The second form is used for a causal
input sequence,in which case y[-1]
is called the initial condition
§ 2.5 Discrete-Time Systems
Linear System
Shift-Invariant System
Causal System
Stable System
Passive and Lossless Systems
§ 2.5.1 Linear Discrete-Time
Systems
Definition - If y1[n] is the output due to an
input x1[n] and y2[n] is the output due to an
input x2[n] then for an input
x[n]=ax1[n]+bx2[n]
the output is given by
y[n]=ay1[n]+by2[n]
Above property must hold for any arbitrary
constants a and b and for all possible inputs
x1[n] and x2[n]
Hence,the above system is linear
§ 2.5.1 Shift-Invariant System
For a shift-invariant system,if y1[n] is the
response to an input x1[n],then the response
to an input
x[n]=x1[n-n0]
is simply
y[n]=y1[n-n0]
where n0 is any positive or negative integer
The above relation must hold for any
arbitrary input and its corresponding output
The above property is called time-invariance
property,or shift-invariant proterty
§ 2.5.2 Linear Time-Invariant system
Linear Time-Invariant (LTI) System -
A system satisfying both the linearity
and the time-invariance property
LTI systems are mathematically easy to
analyze and characterize,and
consequently,easy to design
Highly useful signal processing
algorithms have been developed utilizing
this class of systems over the last several
decades
§ 2.5.3 Passive and 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,i.e.
nn
nxny 22 ][][
For a lossless system,the above inequality is
satisfied with an equal sign for every input
§ 2.5.3 Passive and Lossless
Systems
Example - Consider the discrete-time
system defined by y[n]=?x[n-N] with N a
positive integer
Its output energy is given by
nn
nxny 222 ][][
Hence,it is a passive system if |?|? 1
and is a lossless system if |?| =1
§ 2.5.4 Impulse and Step
Responses
The response of a discrete-time system to a
unit sample sequence {?[n]} is called the unit
impulse response or simply,the impulse
response,and is denoted by {h[n]}
The response of a discrete-time system to a
unit step sequence {?[n]} is called the unit step
response or simply,the step response,and is
denoted by {s[n]}
§ 2.5.4 Impulse and Step
Responses
Example - The impulse response of the
system
y[n]=a1x[n]+a2x[n-1] +a3x[n-2] +a4x[n-3]
is obtained by setting x[n] =?[n]
resulting in
h[n]=a1?[n]+a2?[n-1] +a3?[n-2] +a4?[n-3]
The impulse response is thus a finite-
length sequence of length 4 given by
{h[n]={a1,a2,a3,a4}
§ 2.5.4 Impulse and Step
Responses
Example - The impulse response of the
discrete-time accumulator
n
xny
][][
][][][ nnh
n
is obtained by setting x[n] =?[n] resulting
in
§ 2.5.4 Impulse and Step
Responses
Example - The impulse response {h[n]}
of the factor-of-2 interpolator
])[][(][][ 1121 nxnxnxny uuu
])[][(][][ 1121 nnnnh
}.,.{]}[{ 50150
nh
The impulse response is thus a finite-length
sequence of length 3:
is obtained by setting xu[n]=?[n] and is given by
§ 2.6 Time-Domain Characterization
of LTI Discrete-Time System
Input-Output Relationship -
A consequence of the linear,time-
invariance property is that an LTI
discrete-time system is completely
characterized by its impulse response
Knowing the impulse response one can
compute the output of the system for
any arbitrary input
§ 2.6 Time-Domain Characterization
of LTI Discrete-Time System
Let h[n] denote the impulse response of
a LTI discrete-time system
Compute its output y[n] for the input:
]5[75.0]2[]1[5.1]2[5.0][ nnnnnx
As the system is linear,we can compute
its outputs for each member of the input
separately and add the individual outputs
to determine y[n]
§ 2.6 Time-Domain Characterization
of LTI Discrete-Time System
Likewise,as the system is linear
]2[5.0]2[5.0 nhn
]1[5.1]1[5.1 nhn
]2[]2[ nhn
Input output
][.][.][ 151250 nhnhny
][.][ 57502 nhnh
]5[75.0]5[75.0 nhn
Hence because of the linearity property we get
§ 2.6 Time-Domain Characterization
of LTI Discrete-Time System
Now,any arbitrary input sequence x[n]
can be expressed as a linear combination
of delayed and advanced unit sample
sequences in the form
k
knkxnx ][][][
The response of the LTI system to an
input x[k]?[n-k] will be x[k]h[n-k]
§ 2.6 Time-Domain Characterization
of LTI Discrete-Time System
Hence,the response y[n] to an input
k
knkxnx ][][][
k
knhkxny ][][][
k
khknxny ][][][
which can be alternately written as
will be
§ 2.6.1 Convolution Sum
The summation
kk
nhknxknhkxny ][][][][][
y[n] = x[n] h[n]*
is called the convolution sum of the
sequences x[n] and h[n] and represented
compactly as
§ 2.6.1 Convolution Sum
Properties -
Commutative property:
x[n] h[n] = h[n] x[n]* *
(x[n] h[n]) y[n] = x[n] (h[n] y[n])****
x[n] (h[n] + y[n]) = x[n] h[n] + x[n] y[n]** *
Distributive property,
Associative property,
§ 2.6.1 Convolution Sum
Interpretation -
1) Time-reverse h[k] to form h[-k]
2) Shift h[-k] to the right by n sampling
periods if n > 0 or shift to the left by n
sampling periods if n < 0 to form h[n-k]
3) Form the product v[k]=x[k]h[n-k]
4) Sum all samples of v[k] to develop the n-th
sample of y[n] of the convolution sum
§ 2.6.1 Convolution Sum
Schematic Representation -
nz ][ knh?][ kh?
][kx
][kv ][ny?
k
The computation of an output sample
using the convolution sum is simply a sum
of products
Involves fairly simple operations such as
additions,multiplications,and delays
§ 2.6.1 Convolution Sum
In practice,if either the input or the
impulse response is of finite length,the
convolution sum can be used to compute
the output sample as it involves a finite
sum of products
If both the input sequence and the
impulse response sequence are of finite
length,the output sequence is also of
finite length
§ 2.6.1 Convolution Sum
If both the input sequence and the
impulse response sequence are of infinite
length,convolution sum cannot be used
to compute the output
For systems characterized by an infinite
impulse response sequence,an alternate
time-domain description involving a
finite sum of products will be considered
§ 2.6.1 Convolution Sum
Example,Develop the sequence y[n]
generated by the convolution of the
sequences x[n] and h[n],
x[n] = h[n] = δ[n] + δ[n-1] + δ[n-2]
0
][][][][][
k
knhkxnhnxny *
x[k]
h[k]
x[k]
h[-k]
y[0]
x[k]
h[1-k]
y[1]
x[k]
h[2-k]
y[2]
x[k]
h[3-k]
y[3]
x[k]
h[4-k]
y[4 ]
y[n]=δ[n]+2δ[n-1]+3δ[n-2]+2δ[n-3]+δ[n-4]
x[n] = h[n] = δ[n] + δ[n-1] + δ[n-2]
§ 2.6.1 Convolution Sum
In general,if the lengths of the two
sequences being convolved are M and N,
then the sequence generated by the
convolution is of length M+N-1
§ 2.6.1 Convolution Sum
The M-file conv implements the
convolution sum of two finite-length
sequences
If a=[-2 0 1 -1 3]
b=[1 2 0 -1]
then conv(a,b) yields
[-2 -4 1 3 1 5 1 -3]
§ 2.6.2 Simple Interconnection
Schemes
Cascade Connection
][nh1][nh2][nh1 ][nh2?
][][ nhnh 1? ][2][nh1 *?
Impulse response h[n] of the cascade of two LTI
discrete-time systems with impulse responses
h1[n] and h2[n] is given by
][nh2 ][][ nnh 1? *
§ 2.6.2 Simple Interconnection
Schemes
Note,The ordering of the systems in the
cascade has no effect on the overall
impulse response because of the
commutative property of convolution
A cascade connection of two stable
systems is stable
A cascade connection of two passive
(lossless) systems is passive (lossless)
§ 2.6.2 Simple Interconnection
Schemes
Parallel Connection
][nh2
][nh1
][][ nhnh 1? ][2][nh1
Impulse response h[n] of the parallel
connection of two LTI discrete-time
systems with impulse responses h1[n] and
h2[n] is given by
h[n]= h2[n] + h1[n]
§ 2.6.2 Simple Interconnection
Schemes
h1[n]= δ[n] + 0.5δ[n-1]
h2[n]= 0.5δ[n] + 0.25δ[n-1]
h3[n]= 2δ[n]
h4[n]= 2(0.5)n?[n]
][nh2
][nh1?
][nh4
][nh3
Consider the discrete-time system where
§ 2.6.2 Simple Interconnection
Schemes
Simplifying the block-diagram we obtain
][nh2
][nh1?
][][ 43 nhnh?
][nh1?
])[][(][ 432 nhnhnh?*
])[][(][ 432 nhnhnh?*h1[n]+?
§ 2.6.2 Simple Interconnection
Schemes
Overall impulse response h[n] is given
by
][][][][][ nhnhnhnhnh 42321
])[][(][][][ nhnhnhnhnh 4321*
* *
][2])1[][(][][ 412132 nnnnhnh
]1[][ 21 nn
* *
Now,
§ 2.6.2 Simple Interconnection
Schemes
][][)( 21 nnn
]1[)(][)( 1212121 nn nn
]1[)(][)( 2121 nn nn
][)(2])1[][(][][ 21412142 nnnnhnh n* *
][][]1[][]1[][][ 2121 nnnnnnnh
Therefore
§ 2.7 Classification of LTI
Discrete-Time Systems
Based on Impulse Response Length -
If the impulse response h[n] is of finite length,
i.e.,
h[n]=0 for N1<n<N2 and N1<N2
then it is known as a finite impulse response
(FIR) discrete-time system
The convolution sum description here is
2
1
][][][
N
Nk
knxkhny
§ 2.7 Classification of LTI
Discrete-Time Systems
The output y[n] of an FIR LTI discrete-
time system can be computed directly
from the convolution sum as it is a finite
sum of products
Examples of FIR LTI discrete-time
systems are the moving-average system
and the linear interpolators
§ 2.7 Classification of LTI
Discrete-Time Systems
If the impulse response is of infinite
length,then it is known as an infinite
impulse response (IIR) discrete-time
system
The class of IIR systems we are
concerned with in this course are
characterized by linear constant
coefficient difference equations
§ 2.7 Classification of LTI
Discrete-Time Systems
Example - The discrete-time
accumulator defined by
y[n]=y[n-1]+x[n]
is seen to be an IIR system
§ 2.7 Classification of LTI
Discrete-Time Systems
Example - The familiar numerical
integration formulas that are used to
numerically solve integrals of the form
t
dxty
0
)()(
can be shown to be characterized by linear
constant coefficient difference equations,and
hence,are examples of IIR systems
§ 2.7 Classification of LTI
Discrete-Time Systems
If we divide the interval of integration
into n equal parts of length T,then the
previous integral can be rewritten as
nT
Tn
dxTnynTy
)1(
)())1(()(
nT
dxnTy
0
)()(
where we have set t = nT and used the
notation
§ 2.7 Classification of LTI
Discrete-Time Systems
Using the trapezoidal method we can
write
)}())1(({)(
2
)1(
nTxTnxdx T
nT
Tn
)}())1(({))1(()( 2 nTxTnxTnynTy T
Hence,a numerical representation of the
definite integral is given by
§ 2.7 Classification of LTI
Discrete-Time Systems
Let y[n] = y(nT) and x[n] = x(nT)
Then
)}())1(({))1(()( 2 nTxTnxTnynTy T
]}1[][{]1[][ 2 nxnxnyny T
which is recognized as the difference
equation representation of a first-order
IIR discrete-time system
reduces to
§ 2.7 Classification of LTI
Discrete-Time Systems
Based on the Output Calculation Process
Nonrecursive System - Here the output
can be calculated sequentially,knowing
only the present and past input samples
Recursive System - Here the output
computation involves past output
samples in addition to the present and
past input samples
§ 2.8 Correlation of Signals
There are applications where it is
necessary to compare one reference
signal with one or more signals to
determine the similarity between the
pair and to determine additional
information based on the similarity
§ 2.8 Correlation of Signals
For example,in digital communications,
a set of data symbols are represented by
a set of unique discrete-time sequences
If one of these sequences has been
transmitted,the receiver has to
determine which particular sequence
has been received by comparing the
received signal with every member of
possible sequences from the set
§ 2.8 Correlation of Signals
Similarly,in radar and sonar
applications,the received signal
reflected from the target is a delayed
version of the transmitted signal and by
measuring the delay,one can determine
the location of the target
The detection problem gets more
complicated in practice,as often the
received signal is corrupted by additive
random noise
§ 2.8 Correlation of Signals
Definitions
A measure of similarity between a pair of
energy signals,x[n] and y[n],is given by the
cross-correlation sequence defined by
...,,,],[][][ 210
n
xy nynxr
The parameter l called lag,indicates the time-
shift between the pair of signals
Homework
Read the textbook from p41 to 94
Problems
2.2,2.5,2.8,2.18,2.22,2.23,2.27,2.28,
2.31,2.33,2.37,2.47,2.57,2.58
M2.5,M2.12
Discrete-Time
Signals and Systems
§ 2.1 Discrete-Time Signals:
Time-Domain Representation
Signals represented as sequences of
numbers,called samples
Sample value of a typical signal or
sequence denoted as x[n] with n being an
integer in the range -n
x[n] defined only for integer values of n
and undefined for noninteger values of n
Discrete-time signal represented by {x[n]}
§ 2.1 Discrete-Time Signals:
Time-Domain Representation
Discrete-time signal may also be written
as a sequence of numbers inside braces:
{x[n]}={…,-0.2,2.2,1.1,0.2,-3.7,2.9,…}
The arrow is placed under the sample at
time index n = 0
In the above,x[-1]= -0.2,x[0]=2.2,
x[1]=1.1,etc,
§ 2.1 Discrete-Time Signals:
Time-Domain Representation
Graphical representation of a
discrete-time signal with real-
valued samples is as shown below:
§ 2.1 Discrete-Time Signals:
Time-Domain Representation
In some applications,a discrete-time
sequence {x[n]} may be generated by
periodically sampling a continuous-time
signal xa(t) at uniform intervals of time
§ 2.1 Discrete-Time Signals:
Time-Domain Representation
Here,n-th sample is given by
x[n]=xa(t) |t=nT=xa(nT),n=…,-2,-1,0,1,…
The spacing T between two consecutive
samples is called the sampling interval or
sampling period
Reciprocal of sampling interval T,denoted as
FT,is called the sampling frequency:
FT=1/T
§ 2.1 Discrete-Time Signals:
Time-Domain Representation
Unit of sampling frequency is cycles per
second,or hertz (Hz),if T is in seconds
Whether or not the sequence {x[n]} has been
obtained by sampling,the quantity x[n] is
called the n-th sample of the sequence
{x[n]} is a real sequence,if the n-th sample x[n]
is real for all values of n
Otherwise,{x[n]} is a complex sequence
§ 2.1 Discrete-Time Signals:
Time-Domain Representation
A complex sequence {x[n]} can be
written as {x[n]}={xre[n]}+j{xim[n]}
where xre and xim are the real and
imaginary parts of x[n]
The complex conjugate sequence of {x[n]}
is given by {x*[n]}={xre[n]} - j{xim [n]}
Often the braces are ignored to denote a
sequence if there is no ambiguity
§ 2.1 Discrete-Time Signals:
Time-Domain Representation
Example - {x[n]}={cos0.25n} is a
real sequence {y[n]}={ej0.3n} is a
complex sequence
We can write
{y[n]}={cos0.3n + jsin0.3n}
= {cos0.3n} + j{sin0.3n}
where {yre[n]}={cos0.3n}
{yim[n]}={sin0.3n}
§ 2.1 Discrete-Time Signals:
Time-Domain Representation
Two types of discrete-time signals,
- Sampled-data signals in which samples
are continuous-valued
- Digital signals in which samples are
discrete-valued
Signals in a practical digital signal
processing system are digital signals
obtained by quantizing the sample
values either by rounding or truncation
§ 2.1 Discrete-Time Signals:
Time-Domain Representation
Example -
Am
pli
tud
e
Digital signal
Am
pli
tud
e
Boxedcar signal
Time,t Time,t
§ 2.1 Discrete-Time Signals:
Time-Domain Representation
A discrete-time signal may be a finite-
length or an infinite-length sequence
Finite-length (also called finite-duration
or finite-extent) sequence is defined only
for a finite time interval:N1?n?N2
where -?< N1 and N2 <? with N1? N2
Length or duration of the above finite-
length sequence is N= N2 - N1+ 1
§ 2.1 Discrete-Time Signals:
Time-Domain Representation
Example - x[n]=n2,-3?n? 4 is a
finite-length sequence of length
4 – (-3) + 1 = 8
y[n]=cos0.4n is an infinite-length
sequence
§ 2.1 Discrete-Time Signals:
Time-Domain Representation
A right-sided sequence x[n] has zero-
valued samples for n < N1
A right-sided sequence
If N1? 0,a right-sided sequence is called
a causal sequence
§ 2.2 Operations on Sequences
A single-input,single-output discrete-
time system operates on a sequence,
called the input sequence,according
some prescribed rules and develops
another sequence,called the output
sequence,with more desirable
properties
x[n] y[n]
Input sequence Output sequence
Discrete-time
system
§ 2.2 Operations on Sequences
For example,the input may be a signal
corrupted with additive noise
Discrete-time system is designed to
generate an output by removing the
noise component from the input
In most cases,the operation defining a
particular discrete-time system is
composed of some basic operations
§ 2.2.1 Basic Operations
Product (modulation) operation:
Modulator
x[n] y[n]
w[n]
y[n]=x[n].w[n]
An application is in forming a finite-length
sequence from an infinite-length sequence by
multiplying the latter with a finite-length
sequence called an window sequence
The process is called windowing
§ 2.2.1 Basic Operations
Addition operation:
x[n] y[n]
w[n]
y[n]=x[n]+w[n]–Adder
A
x[n] y[n] y[n]=A.x[n]–Multiplier
Multiplication operation
§ 2.2.1 Basic Operations
Time-shifting operation,where N is an integer
If N > 0,it is delaying operation
1?z y[n]x[n]
–Unit delay
y[n]=x[n-1]
y[n]x[n] z
-Unit advance
y[n]=x[n+1]
If N < 0,it is an advance operation
§ 2.2.1 Basic Operations
Time-reversal (folding) operation:
y[n]=x[-n]
Branching operation,Used to provide
multiple copies of a sequence
x[n] x[n]
x[n]
§ 2.2.1 Basic Operations
Example - Consider the two following
sequences of length 5 defined for 0?n?4,
{a[n]}={3 4 6 –9 0}
{b[n]}={2 –1 4 5 –3}
New sequences generated from the
above two sequences by applying the
basic operations are as follows:
§ 2.2.1 Basic operations
{c[n]}={a[n].b[n]}={6 –4 24 –45 0}
{d[n]}={a[n]+b[n]}={5 3 10 –4 -3}
{e[n]}=(3/2){a[n]}={4.5 6 9 –13.5 0}
As pointed out by the above example,
operations on two or more sequences
can be carried out if all sequences
involved are of same length and defined
for the same range of the time index n
§ 2.2.1 Basic operations
However if the sequences are not of
same length,in some situations,this
problem can be circumvented by
appending zero-valued samples to the
sequence(s) of smaller lengths to make
all sequences have the same range of the
time index
Example - Consider the sequence of
length 3 defined for 0?n? 2,{f[n]}={-2,1,-3}
§ 2.2.1 Basic Operations
We cannot add the length-3 sequence
to the length-5 sequence {a[n]} defined
earlier
We therefore first append {f[n[} with 2
zero-valued samples resulting in a
length-5 sequence {fe[n[}={-2 1 –3 0 0}
Then
{g[n]}={a[n]}+{f[n]}={1 5 3 –9 0}
§ 2.2.1 Basic Operations
Example -
y[n]=?1x[n]+?2x[n-1]+?3[n-2]+?4x[n-3]
§ 2.2.2 Sampling Rate Alteration
Employed to generate a new sequence
y[n] with a sampling rate F’T higher or
lower than that of the sampling rate FT
of a given sequence x[n]
Sampling rate alteration ratio is
R= F’T / FT
If R > 1,the process called interpolation
If R < 1,the process called decimation
§ 2.2.3 Classification of
Sequences based on periodicity
Example -
A sequence satisfying the periodicity
condition is called an periodic sequence
§ 2.2.4 Classification of
Sequences Energy and Power
Signals
Total energy of a sequence x[n] is
defined by
n
nx 2x ][?
An infinite length sequence with finite
sample values may or may not have finite
energy
A finite length sequence with finite
sample values has finite energy
§ 2.2.4 Classification of
Sequences Energy and Power
Signals
The average power of an aperiodic
sequence is defined by
K
KnKK
nxP 212 1x ][lim
K
KnKx
nx 2,][?
Define the energy of a sequence x[n] over
a finite interval -K? n? K as
§ 2.2.4 Classification of
Sequences Energy and Power
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
§ 2.2.4 Classification of
Sequences Energy and Power
Signals
Example - Consider the causal sequence
defined by
00
013
n
nnx n
,
,)(][
Note,x[n] has infinite energy
Its average power is given by
5.412 )1(9lim1912 1lim
0
K
K
KP K
K
nK
x
§ 2.2.4 Classification of Sequences
bounded,absolutely summable and
squaresummable
A sequence x[n] is said to be bounded if
xBnx ][
Example - The sequence x[n]=cos(0.3?n)
is a bounded sequence as
13.0c o s][ nnx
§ 2.2.4 Classification of Sequences
bounded,absolutely summable and
squaresummable
A sequence x[n] is said to be absolutely
summable if
n
nx ][
Example - The sequence
00
030
n
nny n
,
,.][
is an absolutely summable sequence as
4 2 8 5 71301 130
0
...
n
n
§ 2.2.4 Classification of Sequences
bounded,absolutely summable and
squaresummable
A sequence x[n] is said to be square-
summable if
n
nx 2][
Example - The sequence
n
nnh
4.0s i n][
is square-summable but not absolutely
summable
§ 2.3 Basic Sequences
Unit sample sequence -
0,0
0,1][
n
nn?
0,0
0,1][
n
nn?
Unit step sequence -
§ 2.3 Basic Sequences
Real sinusoidal sequence -
x[n]=Acos(?0n+?)
where A is the amplitude,?0 is the angular
frequency,and? is the phase of x[n]
Example -
§ 2.3 Basic Sequences
Exponential sequence -
,][ nAnx n
,)( oo je, jeAA
],[][][ )( nxjnxeeAnx imrenjj oo
),c o s(][ neAnx onre o
)sin (][ neAnx onim o
where
then we can express
If we write
where A and? are real or complex numbers
§ 2.3 Basic Sequences
xre[n] and xim[n] of a complex exponential
sequence are real sinusoidal sequences with
constant (?0=0),growing (?0>0),and decaying
(?0<0) amplitudes for n > 0
njnx )e x p (][ 6121
Real part Imaginary part
§ 2.3 Basic Sequences
Real exponential sequence -
x[n]=A?n,-?< n <?
where A and? are real numbers
=1.2?=0.9
§ 2.3 Basic Sequences
Sinusoidal sequence Acos(?0n +?) and
complex exponential sequence Bexp(j?0n)
are periodic sequences of period N if?0N=2?r
where N and r are positive integers
Smallest value of N satisfying?0N=2?r
is the fundamental period of the sequence
To verify the above fact,consider
x1[n]= Acos(?0n +?)
x2[n]= Acos(?0 ( n+N) +?)
§ 2.3 Basic Sequences
Now
x2[n]= cos(?0 ( n+N) +?)
= cos(?0n+?)cos?0N - sin(?0n+?)sin?0N
which will be equal to cos(?0n+?)=x1[n]
only if sin?0N= 0 and cos?0N = 1
These two conditions are met if and only
if?0N= 2?r or 2?/?0 = N/r
§ 2.3 Basic Sequences
If 2?/?0 is a noninteger rational number,
then the period will be a multiple of
2?/?0
Otherwise,the sequence is aperiodic
Example - x[n]=sin(?3n+?) is an
aperiodic sequence
§ 2.3 Basic Sequences
Here?0 = 0.1?
Hence N= 2?r/?0 = 20 for r = 1
0 = 0.1?
§ 2.3 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
]2[]1[5.1]2[5.0][ nnnnx ]6[75.0]4[ nn
§ 2.4 The Sampling Process
Often,a discrete-time sequence x[n] is
developed by uniformly sampling a
continuous-time signal xa(t) as indicated below
The relation between the two signals is
x[n]=xa(t)|t=nT=xa(nT),n=…,-2,-1,0,1,2,…
§ 2.4 The Sampling Process
Time variable t of xa(t) is related to the time
variable n of x[n] only at discrete-time instants
tn given by
tn= nT = n/FT = 2?n/?T
with FT=1/T denoting the sampling frequency
and?T= 2?FT denoting the sampling angular
frequency
§ 2.4 The Sampling Process
Consider the continuous-time signal
)c o s ()2c o s ()( tAtfAtx oo
)2c o s()c o s(][ nAnTAnx
T
oo
)c o s ( nA o
ToToo /2
is the normalized digital angular frequency of
x[n]
where
The corresponding discrete-time signal is
§ 2.4 The Sampling Process
If the unit of sampling period T is in
seconds
The unit of normalized digital angular
frequency?0 is radians/sample
The unit of normalized analog angular
frequency?0 is radians/second
The unit of analog frequency f0 is hertz
(Hz)
§ 2.4 The Sampling Process
The three continuous-time signals
)6c o s ()(1 ttg
)14c o s ()(2 ttg
)26c o s ()(3 ttg
)6.0c o s (][1 nng
)4.1c o s (][2 nng
)6.2c o s (][3 nng
of frequencies 3 Hz,7 Hz,and 13 Hz,are
sampled at a sampling rate of 10 Hz,i.e,with T =
0.1 sec,generating the three sequences
§ 2.4 The Sampling Process
Plots of these sequences (shown with circles)
and their parent time functions are shown
below:
Note that each sequence has exactly the same
sample value for any given n
0 0.2 0.4 0.6 0.8 1
-1
- 0.5
0
0.5
1
t i m e
A
m
pl
i
t
ude
§ 2.4 The Sampling Process
This fact can also be verified by observing that
)6.0c o s ()6.02(c o s)4.1c o s (][2 nnnng
)6.0c o s ()6.02(c o s)6.2c o s (][3 nnnng
As a result,all three sequences are identical
and it is difficult to associate a unique
continuous-time function with each of these
sequences
§ 2.4 The Sampling Process
The above phenomenon of a continuous-
time signal of higher frequency
acquiring the identity of a sinusoidal
sequence of lower frequency after
sampling is called aliasing
§ 2.4 The Sampling Process
Recall
0=20/?T
Thus if?T>2?0,then the corresponding
normalized digital angular frequency?0 of the
discrete-time signal obtained by sampling the
parent continuous-time sinusoidal signal will
be in the range -?<?<?
Conclusion,No aliasing
§ 2.4 The Sampling Process
On the other hand,if?T < 2?0,the
normalized digital angular frequency will
foldover into a lower digital frequency
0 = ( 20 /?T)2? in the range -?<?<?
because of aliasing
Hence,to prevent aliasing,the sampling
frequency?T should be greater than 2 times
the frequency?0 of the sinusoidal signal being
sampled
§ 2.4 The Sampling Process
Generalization,Consider an arbitrary
continuous-time signal xa(t) composed of a
weighted sum of a number of sinusoidal
signals
xa(t) can be represented uniquely by its
sampled version {x[n]} if the sampling
frequency?T is chosen to be greater than 2
times the highest frequency contained in xa(t)
The condition to be satisfied by the sampling
frequency to prevent aliasing is called the
sampling theorem
§ 2.5 Discrete-Time Systems
A discrete-time system processes a given
input sequence x[n] to generates an
output sequence y[n] with more
desirable properties
In most applications,the discrete-time
system is a single-input,single-output
system:
x[n] y[n]
Input sequence Output sequence
Discrete-Time
System
§ 2.5 Discrete-Time Systems
2-input,1-output discrete-time systems -
Modulator,adder
1-input,1-output discrete-time systems -
Multiplier,unit delay,unit advance
§ 2.5 Discrete-Time Systems
Accumulator,
n xny
][][
][]1[][][1 nxnynxxn
The output y[n] at time instant n is the sum of
the input sample x[n] at time instant n and the
previous output y[n-1] at time instant n-1
which is the sum of all previous input sample
values from -? to n-1
The system accumulatively adds,i.e.,it
accumulates all input sample values
§ 2.5 Discrete-Time Systems
Accumulator - Input-output relation can
also be written in the form
n xxny
0
1 ][][][
,][]1[
0
n xy
0?n
The second form is used for a causal
input sequence,in which case y[-1]
is called the initial condition
§ 2.5 Discrete-Time Systems
Linear System
Shift-Invariant System
Causal System
Stable System
Passive and Lossless Systems
§ 2.5.1 Linear Discrete-Time
Systems
Definition - If y1[n] is the output due to an
input x1[n] and y2[n] is the output due to an
input x2[n] then for an input
x[n]=ax1[n]+bx2[n]
the output is given by
y[n]=ay1[n]+by2[n]
Above property must hold for any arbitrary
constants a and b and for all possible inputs
x1[n] and x2[n]
Hence,the above system is linear
§ 2.5.1 Shift-Invariant System
For a shift-invariant system,if y1[n] is the
response to an input x1[n],then the response
to an input
x[n]=x1[n-n0]
is simply
y[n]=y1[n-n0]
where n0 is any positive or negative integer
The above relation must hold for any
arbitrary input and its corresponding output
The above property is called time-invariance
property,or shift-invariant proterty
§ 2.5.2 Linear Time-Invariant system
Linear Time-Invariant (LTI) System -
A system satisfying both the linearity
and the time-invariance property
LTI systems are mathematically easy to
analyze and characterize,and
consequently,easy to design
Highly useful signal processing
algorithms have been developed utilizing
this class of systems over the last several
decades
§ 2.5.3 Passive and 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,i.e.
nn
nxny 22 ][][
For a lossless system,the above inequality is
satisfied with an equal sign for every input
§ 2.5.3 Passive and Lossless
Systems
Example - Consider the discrete-time
system defined by y[n]=?x[n-N] with N a
positive integer
Its output energy is given by
nn
nxny 222 ][][
Hence,it is a passive system if |?|? 1
and is a lossless system if |?| =1
§ 2.5.4 Impulse and Step
Responses
The response of a discrete-time system to a
unit sample sequence {?[n]} is called the unit
impulse response or simply,the impulse
response,and is denoted by {h[n]}
The response of a discrete-time system to a
unit step sequence {?[n]} is called the unit step
response or simply,the step response,and is
denoted by {s[n]}
§ 2.5.4 Impulse and Step
Responses
Example - The impulse response of the
system
y[n]=a1x[n]+a2x[n-1] +a3x[n-2] +a4x[n-3]
is obtained by setting x[n] =?[n]
resulting in
h[n]=a1?[n]+a2?[n-1] +a3?[n-2] +a4?[n-3]
The impulse response is thus a finite-
length sequence of length 4 given by
{h[n]={a1,a2,a3,a4}
§ 2.5.4 Impulse and Step
Responses
Example - The impulse response of the
discrete-time accumulator
n
xny
][][
][][][ nnh
n
is obtained by setting x[n] =?[n] resulting
in
§ 2.5.4 Impulse and Step
Responses
Example - The impulse response {h[n]}
of the factor-of-2 interpolator
])[][(][][ 1121 nxnxnxny uuu
])[][(][][ 1121 nnnnh
}.,.{]}[{ 50150
nh
The impulse response is thus a finite-length
sequence of length 3:
is obtained by setting xu[n]=?[n] and is given by
§ 2.6 Time-Domain Characterization
of LTI Discrete-Time System
Input-Output Relationship -
A consequence of the linear,time-
invariance property is that an LTI
discrete-time system is completely
characterized by its impulse response
Knowing the impulse response one can
compute the output of the system for
any arbitrary input
§ 2.6 Time-Domain Characterization
of LTI Discrete-Time System
Let h[n] denote the impulse response of
a LTI discrete-time system
Compute its output y[n] for the input:
]5[75.0]2[]1[5.1]2[5.0][ nnnnnx
As the system is linear,we can compute
its outputs for each member of the input
separately and add the individual outputs
to determine y[n]
§ 2.6 Time-Domain Characterization
of LTI Discrete-Time System
Likewise,as the system is linear
]2[5.0]2[5.0 nhn
]1[5.1]1[5.1 nhn
]2[]2[ nhn
Input output
][.][.][ 151250 nhnhny
][.][ 57502 nhnh
]5[75.0]5[75.0 nhn
Hence because of the linearity property we get
§ 2.6 Time-Domain Characterization
of LTI Discrete-Time System
Now,any arbitrary input sequence x[n]
can be expressed as a linear combination
of delayed and advanced unit sample
sequences in the form
k
knkxnx ][][][
The response of the LTI system to an
input x[k]?[n-k] will be x[k]h[n-k]
§ 2.6 Time-Domain Characterization
of LTI Discrete-Time System
Hence,the response y[n] to an input
k
knkxnx ][][][
k
knhkxny ][][][
k
khknxny ][][][
which can be alternately written as
will be
§ 2.6.1 Convolution Sum
The summation
kk
nhknxknhkxny ][][][][][
y[n] = x[n] h[n]*
is called the convolution sum of the
sequences x[n] and h[n] and represented
compactly as
§ 2.6.1 Convolution Sum
Properties -
Commutative property:
x[n] h[n] = h[n] x[n]* *
(x[n] h[n]) y[n] = x[n] (h[n] y[n])****
x[n] (h[n] + y[n]) = x[n] h[n] + x[n] y[n]** *
Distributive property,
Associative property,
§ 2.6.1 Convolution Sum
Interpretation -
1) Time-reverse h[k] to form h[-k]
2) Shift h[-k] to the right by n sampling
periods if n > 0 or shift to the left by n
sampling periods if n < 0 to form h[n-k]
3) Form the product v[k]=x[k]h[n-k]
4) Sum all samples of v[k] to develop the n-th
sample of y[n] of the convolution sum
§ 2.6.1 Convolution Sum
Schematic Representation -
nz ][ knh?][ kh?
][kx
][kv ][ny?
k
The computation of an output sample
using the convolution sum is simply a sum
of products
Involves fairly simple operations such as
additions,multiplications,and delays
§ 2.6.1 Convolution Sum
In practice,if either the input or the
impulse response is of finite length,the
convolution sum can be used to compute
the output sample as it involves a finite
sum of products
If both the input sequence and the
impulse response sequence are of finite
length,the output sequence is also of
finite length
§ 2.6.1 Convolution Sum
If both the input sequence and the
impulse response sequence are of infinite
length,convolution sum cannot be used
to compute the output
For systems characterized by an infinite
impulse response sequence,an alternate
time-domain description involving a
finite sum of products will be considered
§ 2.6.1 Convolution Sum
Example,Develop the sequence y[n]
generated by the convolution of the
sequences x[n] and h[n],
x[n] = h[n] = δ[n] + δ[n-1] + δ[n-2]
0
][][][][][
k
knhkxnhnxny *
x[k]
h[k]
x[k]
h[-k]
y[0]
x[k]
h[1-k]
y[1]
x[k]
h[2-k]
y[2]
x[k]
h[3-k]
y[3]
x[k]
h[4-k]
y[4 ]
y[n]=δ[n]+2δ[n-1]+3δ[n-2]+2δ[n-3]+δ[n-4]
x[n] = h[n] = δ[n] + δ[n-1] + δ[n-2]
§ 2.6.1 Convolution Sum
In general,if the lengths of the two
sequences being convolved are M and N,
then the sequence generated by the
convolution is of length M+N-1
§ 2.6.1 Convolution Sum
The M-file conv implements the
convolution sum of two finite-length
sequences
If a=[-2 0 1 -1 3]
b=[1 2 0 -1]
then conv(a,b) yields
[-2 -4 1 3 1 5 1 -3]
§ 2.6.2 Simple Interconnection
Schemes
Cascade Connection
][nh1][nh2][nh1 ][nh2?
][][ nhnh 1? ][2][nh1 *?
Impulse response h[n] of the cascade of two LTI
discrete-time systems with impulse responses
h1[n] and h2[n] is given by
][nh2 ][][ nnh 1? *
§ 2.6.2 Simple Interconnection
Schemes
Note,The ordering of the systems in the
cascade has no effect on the overall
impulse response because of the
commutative property of convolution
A cascade connection of two stable
systems is stable
A cascade connection of two passive
(lossless) systems is passive (lossless)
§ 2.6.2 Simple Interconnection
Schemes
Parallel Connection
][nh2
][nh1
][][ nhnh 1? ][2][nh1
Impulse response h[n] of the parallel
connection of two LTI discrete-time
systems with impulse responses h1[n] and
h2[n] is given by
h[n]= h2[n] + h1[n]
§ 2.6.2 Simple Interconnection
Schemes
h1[n]= δ[n] + 0.5δ[n-1]
h2[n]= 0.5δ[n] + 0.25δ[n-1]
h3[n]= 2δ[n]
h4[n]= 2(0.5)n?[n]
][nh2
][nh1?
][nh4
][nh3
Consider the discrete-time system where
§ 2.6.2 Simple Interconnection
Schemes
Simplifying the block-diagram we obtain
][nh2
][nh1?
][][ 43 nhnh?
][nh1?
])[][(][ 432 nhnhnh?*
])[][(][ 432 nhnhnh?*h1[n]+?
§ 2.6.2 Simple Interconnection
Schemes
Overall impulse response h[n] is given
by
][][][][][ nhnhnhnhnh 42321
])[][(][][][ nhnhnhnhnh 4321*
* *
][2])1[][(][][ 412132 nnnnhnh
]1[][ 21 nn
* *
Now,
§ 2.6.2 Simple Interconnection
Schemes
][][)( 21 nnn
]1[)(][)( 1212121 nn nn
]1[)(][)( 2121 nn nn
][)(2])1[][(][][ 21412142 nnnnhnh n* *
][][]1[][]1[][][ 2121 nnnnnnnh
Therefore
§ 2.7 Classification of LTI
Discrete-Time Systems
Based on Impulse Response Length -
If the impulse response h[n] is of finite length,
i.e.,
h[n]=0 for N1<n<N2 and N1<N2
then it is known as a finite impulse response
(FIR) discrete-time system
The convolution sum description here is
2
1
][][][
N
Nk
knxkhny
§ 2.7 Classification of LTI
Discrete-Time Systems
The output y[n] of an FIR LTI discrete-
time system can be computed directly
from the convolution sum as it is a finite
sum of products
Examples of FIR LTI discrete-time
systems are the moving-average system
and the linear interpolators
§ 2.7 Classification of LTI
Discrete-Time Systems
If the impulse response is of infinite
length,then it is known as an infinite
impulse response (IIR) discrete-time
system
The class of IIR systems we are
concerned with in this course are
characterized by linear constant
coefficient difference equations
§ 2.7 Classification of LTI
Discrete-Time Systems
Example - The discrete-time
accumulator defined by
y[n]=y[n-1]+x[n]
is seen to be an IIR system
§ 2.7 Classification of LTI
Discrete-Time Systems
Example - The familiar numerical
integration formulas that are used to
numerically solve integrals of the form
t
dxty
0
)()(
can be shown to be characterized by linear
constant coefficient difference equations,and
hence,are examples of IIR systems
§ 2.7 Classification of LTI
Discrete-Time Systems
If we divide the interval of integration
into n equal parts of length T,then the
previous integral can be rewritten as
nT
Tn
dxTnynTy
)1(
)())1(()(
nT
dxnTy
0
)()(
where we have set t = nT and used the
notation
§ 2.7 Classification of LTI
Discrete-Time Systems
Using the trapezoidal method we can
write
)}())1(({)(
2
)1(
nTxTnxdx T
nT
Tn
)}())1(({))1(()( 2 nTxTnxTnynTy T
Hence,a numerical representation of the
definite integral is given by
§ 2.7 Classification of LTI
Discrete-Time Systems
Let y[n] = y(nT) and x[n] = x(nT)
Then
)}())1(({))1(()( 2 nTxTnxTnynTy T
]}1[][{]1[][ 2 nxnxnyny T
which is recognized as the difference
equation representation of a first-order
IIR discrete-time system
reduces to
§ 2.7 Classification of LTI
Discrete-Time Systems
Based on the Output Calculation Process
Nonrecursive System - Here the output
can be calculated sequentially,knowing
only the present and past input samples
Recursive System - Here the output
computation involves past output
samples in addition to the present and
past input samples
§ 2.8 Correlation of Signals
There are applications where it is
necessary to compare one reference
signal with one or more signals to
determine the similarity between the
pair and to determine additional
information based on the similarity
§ 2.8 Correlation of Signals
For example,in digital communications,
a set of data symbols are represented by
a set of unique discrete-time sequences
If one of these sequences has been
transmitted,the receiver has to
determine which particular sequence
has been received by comparing the
received signal with every member of
possible sequences from the set
§ 2.8 Correlation of Signals
Similarly,in radar and sonar
applications,the received signal
reflected from the target is a delayed
version of the transmitted signal and by
measuring the delay,one can determine
the location of the target
The detection problem gets more
complicated in practice,as often the
received signal is corrupted by additive
random noise
§ 2.8 Correlation of Signals
Definitions
A measure of similarity between a pair of
energy signals,x[n] and y[n],is given by the
cross-correlation sequence defined by
...,,,],[][][ 210
n
xy nynxr
The parameter l called lag,indicates the time-
shift between the pair of signals
Homework
Read the textbook from p41 to 94
Problems
2.2,2.5,2.8,2.18,2.22,2.23,2.27,2.28,
2.31,2.33,2.37,2.47,2.57,2.58
M2.5,M2.12
