Unformatted text preview: DiscreteTime Systems:
Examples DiscreteTime Systems
• A discretetime system processes a given
input sequence x[n] to generates an output
sequence y[n] with more desirable
properties
• In most applications, the discretetime
system is a singleinput, singleoutput
system:
Discrete − time
System x [n ]
1 Input sequence • 2input, 1output discretetime systems Modulator, adder
• 1input, 1output discretetime systems Multiplier, unit delay, unit advance y [n ]
Output sequence 2 Copyright © 2005, S. K. Mitra Copyright © 2005, S. K. Mitra DiscreteTime Systems: Examples
Discrete DiscreteTime Systems:Examples
Discrete n • Accumulator  y[n] = ∑ x[l] • Accumulator  Inputoutput relation can
also be written in the form l = −∞ n −1 = ∑ x[l] + x[ n] = y[n − 1] + x[n] −1 3 • 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 cumulatively adds, i.e., it
accumulates all input sample values l = −∞ l =0
n = y[−1] + ∑ x[l], n ≥ 0
l=0 • The second form is used for a causal input
sequence, in which case y[−1] is called
the initial condition
4 Copyright © 2005, S. K. Mitra Copyright © 2005, S. K. Mitra DiscreteTime Systems:Examples
Discrete DiscreteTime Systems:Examples
Discrete • Mpoint movingaverage system  • If there is no bias in the measurements, an
improved estimate of the noisy data is
obtained by simply increasing M
• A direct implementation of the Mpoint
moving average system requires M − 1
additions, 1 division, and storage of M − 1
past input data samples
• A more efficient implementation is
developed next M −1 1
y[ n] = M ∑ x[ n − k ]
k =0 • Used in smoothing random variations in
data
• In most applications, the data x[n] is a
bounded sequence
•
Mpoint average y[n] is also a
bounded sequence
5 n y[n] = ∑ x[l] + ∑ x[l] l = −∞ 6
Copyright © 2005, S. K. Mitra Copyright © 2005, S. K. Mitra 1 DiscreteTime Systems:Examples
Discrete DiscreteTime Systems:Examples
Discrete 1 ⎛ M −1
⎜ ∑ x[ n − l] +
M⎝
l =0
1⎛M ⎞
x[ n − M ] − x[ n − M ] ⎟
⎠
⎞
= ⎜ ∑ x[ n − l] + x[ n] − x[ n − M ] ⎟
M⎝
⎠
l =1
M −1
⎞
1⎛
= ⎜ ∑ x[ n − 1 − l] + x[ n] − x[ n − M ] ⎟
M⎝
⎠
l =0
Hence
y[ n] = y[ n] = y[ n − 1] + • Computation of the modified Mpoint
moving average system using the recursive
equation now requires 2 additions and 1
division
• An application: Consider
x[n] = s[n] + d[n],
where s[n] is the signal corrupted by a noise
d[n] 1
( x[ n] − x[ n − M ])
M 7 8
Copyright © 2005, S. K. Mitra Copyright © 2005, S. K. Mitra DiscreteTime Systems:Examples
Discrete DiscreteTime Systems:Examples
Discrete s[ n] = 2[n(0.9) ], d[n]  random signal
n • Exponentially Weighted Running Average
Filter
y[ n] = αy[ n − 1] + x[ n], 0 < α < 1 8
d[n]
s[n]
x[n] Amplitude 6
4
2 • Computation of the running average requires
only 2 additions, 1 multiplication and storage
of the previous running average
• Does not require storage of past input data
samples 0
2
0 10 20
30
Time index n 40 50 7
s[n]
y[n] 6 Amplitude 5
4
3
2
1 9 0
0 10 20
30
Time index n 40 50 10 Copyright © 2005, S. K. Mitra Copyright © 2005, S. K. Mitra DiscreteTime Systems:Examples
Discrete DiscreteTime Systems:Examples
Discrete• Linear interpolation  Employed to estimate
sample values between pairs of adjacent
sample values of a discretetime sequence
• Factorof4 interpolation • For 0 < α < 1, the exponentially weighted
average filter places more emphasis on current
data samples and less emphasis on past data
samples as illustrated below
y[ n] = α(αy[ n − 2] + x[ n − 1]) + x[ n]
= α 2 y[ n − 2] + αx[ n − 1] + x[ n]
= α 2 (αy[ n − 3] + x[ n − 2]) + αx[ n − 1] + x[ n] y[n] = α3 y[ n − 3] + α 2 x[ n − 2] + αx[ n − 1] + x[ n]
3 11 12
Copyright © 2005, S. K. Mitra 0 1 2 4
5 6 7 8 9 10 11 12 n Copyright © 2005, S. K. Mitra 2 DiscreteTime Systems:
Examples DiscreteTime Systems:
Examples • Factorof2 interpolator  • Factorof2 interpolator y[n] = xu [ n] + 1 ( xu [n − 1] + xu [n + 1])
2 • Factorof3 interpolator y[n] = xu [n] + 1 ( xu [n − 1] + xu [ n + 2])
3
+ 2 ( xu [n − 2] + xu [n + 1])
3
13 Original ( 512×512 ) Down −sampled
( 256×256 ) Interpolated ( 512 × 512 ) 14
Copyright © 2005, S. K. Mitra Copyright © 2005, S. K. Mitra DiscreteTime Systems:
Examples DiscreteTime Systems:
Examples Median Filter –
• 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 rankordering
the numbers in the set by their values and
choosing the number at the middle
15 Median Filter –
• Example: Consider the set of numbers {2, − 3, 10, 5, − 1}
• Rankorder set is given by
{− 3, − 1, 2, 5, 10}
• Hence, med{2, − 3, 10, 5, − 1} = 2
16 Copyright © 2005, S. K. Mitra Copyright © 2005, S. K. Mitra DiscreteTime Systems:
Examples DiscreteTime Systems:
Examples Median Filter –
• Implemented by sliding a window of odd
length over the input sequence {x[n]} one
sample at a time
• Output y[n] at instant n is the median value
of the samples inside the window centered
at n
17 Median Filter –
• Finds applications in removing additive
random noise, which shows up as sudden
large errors in the corrupted signal
• Usually used for the smoothing of signals
corrupted by impulse noise
18 Copyright © 2005, S. K. Mitra Copyright © 2005, S. K. Mitra 3 DiscreteTime Systems:
Examples DiscreteTime Systems:
Classification Median Filtering Example – •
•
•
•
• 19 Linear System
ShiftInvariant System
Causal System
Stable System
Passive and Lossless Systems 20
Copyright © 2005, S. K. Mitra Copyright © 2005, S. K. Mitra Linear DiscreteTime Systems Linear DiscreteTime Systems
n • 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] = α x1[n] + β x2 [n]
the output is given by
y[n] = α y1[n] + β y2 [n] 21 • Above property must hold for any arbitrary
constants α and β , and for all possible
inputs x1[n] and x2 [n] n • Accumulator  y1[n] = ∑ x1[l], y2 [n] = ∑ x2 [l]
l = −∞
l = −∞
For an input
x[n] = α x1[n] + β x2 [n]
the output is
n y[n] = ∑ (α x1[l] + β x2 [l])
l = −∞
n n l = −∞ l = −∞ = α ∑ x1[l] + β ∑ x2 [l] = α y1[n] + β y2 [n]
• Hence, the above system is linear
22 Copyright © 2005, S. K. Mitra Copyright © 2005, S. K. Mitra Linear DiscreteTime Systems Linear DiscreteTime Systems • The outputs y1[n] and y2 [n] for inputs x1[n]
and x2 [n] are given by n
y1[n] = y1[−1] + ∑ x1[l]
y2 [n] = y2 [−1] + • Now α y1[n] + β y2 [n]
= α ( y1[−1] + l =0
n ∑ x2 [l] y[n] = y[−1] + ∑ (α x1[l] + β x 2 [l])
Copyright © 2005, S. K. Mitra l =0
n n l =0 ∑ x2 [l]) l =0 • Thus y[n] = α y1[n] + β y2 [n] if n l =0 l =0 = (α y1[−1] + β y2 [−1]) + (α ∑ x1[l] + β l =0 • The output y[n] for an input α x1[n] + β x 2 [n]
is given by
23 n n ∑ x1[l]) + β ( y2 [−1] + ∑ x2 [l]) y[−1] = α y1[−1] + β y2 [−1]
24
Copyright © 2005, S. K. Mitra 4 Nonlinear DiscreteTime
System Linear DiscreteTime System 25 • For the causal accumulator to be linear the
condition y[−1] = α y1[−1] + β y2 [−1]
must hold for all initial conditions y[−1],
y1[−1] , y2 [−1] , and all constants α and β
• This condition cannot be satisfied unless the
accumulator is initially at rest with zero
initial condition
• For nonzero initial condition, the system is
nonlinear 26 • The median filter described earlier is a
nonlinear discretetime system
• To show this, consider a median filter with
a window of length 3
• Output of the filter for an input
{x1[ n]} = {3, 4, 5}, 0 ≤ n ≤ 2
is
{y1[ n]} = {3, 4, 4}, 0 ≤ n ≤ 2 Copyright © 2005, S. K. Mitra Copyright © 2005, S. K. Mitra Nonlinear DiscreteTime
System Nonlinear DiscreteTime
System • Output for an input {x2 [ n]} = {2, − 1, − 1}, 0 ≤ n ≤ 2
is • Note {y1[ n] + y2 [ n]} = {3, 3, 3} ≠ {y[ n]} {y2 [ n]} = {0, − 1, − 1}, 0 ≤ n ≤ 2 • Hence, the median filter is a nonlinear
discretetime system • However, the output for an input {x[ n]} = {x1[ n] + x2 [ n]}
is {y[ n]} = {3, 4, 3} 27 28
Copyright © 2005, S. K. Mitra Copyright © 2005, S. K. Mitra ShiftInvariant System ShiftInvariant System • For a shiftinvariant system, if y1[n] is the
response to an input x1[n], then the response
to an input
x[n] = x1[n − no ]
is simply
y[n] = y1[n − no ] 29 where no is any positive or negative integer
• The above relation must hold for any
arbitrary input and its corresponding output
Copyright © 2005, S. K. Mitra • In the case of sequences and systems with
indices n related to discrete instants of time,
the above property is called timeinvariance
property
• Timeinvariance property ensures that for a
specified input, the output is independent of
the time the input is being applied
30
Copyright © 2005, S. K. Mitra 5 ShiftInvariant System ShiftInvariant System • Example  Consider the upsampler with an
inputoutput relation given by
x[n / L], n = 0, ± L, ± 2 L, .....
xu [n] = ⎧
⎨ 0,
otherwise
⎩ 31 • For an input x1[n] = x[n − no ] the output x1,u [n]
is given by
x [n / L], n = 0, ± L, ± 2 L, .....
x1,u [n] = ⎧ 1
⎨ 0,
otherwise
⎩
x[(n − Lno ) / L], n = 0, ± L, ± 2 L, .....
=⎧
⎨
0,
otherwise
⎩
Copyright © 2005, S. K. Mitra • However from the definition of the upsampler
xu [n − no ]
x[(n − no ) / L], n = no , no ± L, no ± 2 L, .....
=⎧
⎨
0,
otherwise
⎩
≠ x1,u [n]
• Hence, the upsampler is a timevarying system
32
Copyright © 2005, S. K. Mitra Linear TimeInvariant System Causal System • Linear TimeInvariant (LTI) System A system satisfying both the linearity and
the timeinvariance 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 • In a causal system, the no th output sample
y[no ] depends only on input samples x[n]
for n ≤ no and does not depend on input
samples for n > no
• Let y1[n] and y2 [n] be the responses of a
causal discretetime system to the inputs x1[n]
and x2 [n] , respectively 33 34
Copyright © 2005, S. K. Mitra Copyright © 2005, S. K. Mitra Causal System Causal System • Examples of causal systems:
y[n] = α1x[n] + α 2 x[n − 1] + α 3 x[n − 2] + α 4 x[n − 3] • Then y[n] = b0 x[n] + b1x[n − 1] + b2 x[n − 2]
+ a1 y[n − 1] + a2 y[n − 2]
y[n] = y[n − 1] + x[n] x1[n] = x2[n] for n < N
implies also that
y1[n] = y2[n] for n < N
• For a causal system, changes in output
samples do not precede changes in the input
samples
35 • Examples of noncausal systems:
1
y[n] = xu [n] + ( xu [n − 1] + xu [n + 1])
2
1 36
Copyright © 2005, S. K. Mitra y[n] = xu [n] + ( xu [n − 1] + xu [n + 2])
3
2
+ ( xu [n − 2] + xu [n + 1])
3 Copyright © 2005, S. K. Mitra 6 Stable System Causal System • There are various definitions of stability
• We consider here the boundedinput,
boundedoutput (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] ≤ B y for all values of n • A noncausal system can be implemented as
a causal system by delaying the output by
an appropriate number of samples
• For example a causal implementation of the
factorof2 interpolator is given by
1
2 y[n] = xu [n − 1] + ( xu [n − 2] + xu [n])
37 38
Copyright © 2005, S. K. Mitra Copyright © 2005, S. K. Mitra Stable System Passive and Lossless Systems • Example  The Mpoint moving average
filter is BIBO stable: y[n] = 1
M • A discretetime system is defined to be
passive if, for every finiteenergy input x[n],
the output y[n] has, at most, the same energy,
i.e. M −1 ∑ x[n − k ] k =0 • For a bounded input x[n] ≤ Bx we have y[n] =
≤ 1
M
1
M M −1 ∑ x[n − k ] ≤ k =0 1
M ∞ n = −∞ ∑ x[n − k ] k =0 40
Copyright © 2005, S. K. Mitra Copyright © 2005, S. K. Mitra Impulse and Step Responses Passive and Lossless Systems • The response of a discretetime system to a
unit sample sequence {δ[n]} is called the
unit sample response or simply, the
impulse response, and is denoted by {h[n]}
• The response of a discretetime system to a
unit step sequence {µ[n]} is called the unit
step response or simply, the step response,
and is denoted by {s[n]} • Example  Consider the discretetime
system defined by y[n] = α x[n − N ] with N
a positive integer
• Its output energy is given by
∞ 2 ∑ y[n] = α n = −∞ 41 n = −∞ • For a lossless system, the above inequality is
satisfied with an equal sign for every input ( MBx ) ≤ Bx 39 ∞ 2
2
∑ y[n] ≤ ∑ x[n] < ∞ M −1 2∞ ∑ x[n] 2 n = −∞ • Hence, it is a passive system if α < 1 and is
a lossless system if α = 1
Copyright © 2005, S. K. Mitra 42
Copyright © 2005, S. K. Mitra 7 Impulse Response Impulse Response • Example  The impulse response of the
system
y[n] = α1x[n] + α 2 x[n − 1] + α 3 x[n − 2] + α 4 x[n − 3] 43 is obtained by setting x[n] = δ[n] resulting
in
h[n] = α1δ [n] + α 2δ [n − 1] + α 3δ [n − 2] + α 4δ [n − 3]
• The impulse response is thus a finitelength
sequence of length 4 given by
{h[n]} = {α1, α 2 , α 3 , α 4}
↑ • Example  The impulse response of the
discretetime accumulator
y[n] = l = −∞ is obtained by setting x[n] = δ[n] resulting
in
n
h[n] = ∑ δ [l] = µ [n]
l = −∞ 44 Copyright © 2005, S. K. Mitra Copyright © 2005, S. K. Mitra TimeDomain Characterization
of LTI DiscreteTime System Impulse Response
• Example  The impulse response {h[n]} of
the factorof2 interpolator
1
y[n] = xu [n] + ( xu [n − 1] + xu [n + 1])
2
• is obtained by setting xu [n] = δ [n] and is
given by
1
h[n] = δ [n] + (δ [n − 1] + δ [n + 1]) • InputOutput Relationship A consequence of the linear, timeinvariance property is that an LTI discretetime 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 45 n ∑ x[l] • The impulse response is thus a finitelength
sequence of length 3:
{h[n]} = {0.5, 1 0.5}
↑ 46 Copyright © 2005, S. K. Mitra Copyright © 2005, S. K. Mitra TimeDomain Characterization
of LTI DiscreteTime System TimeDomain Characterization
of LTI DiscreteTime System • Let h[n] denote the impulse response of a
LTI discretetime system
• We compute its output y[n] for the input: • Since the system is timeinvariant
input output δ[n + 2] → h[n + 2] x[ n] = 0.5δ[ n + 2] + 1.5δ[n − 1] − δ[ n − 2] + 0.75δ[ n − 5] δ[n − 1] → h[n − 1] • 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] δ[n − 2] → h[n − 2] 47 δ[n − 5] → h[n − 5]
48 Copyright © 2005, S. K. Mitra Copyright © 2005, S. K. Mitra 8 TimeDomain Characterization
of LTI DiscreteTime System TimeDomain Characterization
of LTI DiscreteTime System • Likewise, as the system is linear
input 49 • Now, any arbitrary input sequence x[n] can
be expressed as a linear combination of
delayed and advanced unit sample
sequences in the form output 0.5δ[n + 2] → 0.5h[n + 2]
1.5δ[n − 1] → 1.5h[n − 1]
− δ[n − 2] → − h[n − 2]
0.75δ[n − 5] → 0.75h[n − 5]
• Hence because of the linearity property we
get
y[n] = 0.5h[n + 2] + 1.5h[n − 1]
− h[n − 2] + 0.75h[n − 5] ∞ x[n] = ∑ x[k ] δ[n − k ]
k = −∞ • The response of the LTI system to an input
x[ k ] δ[n − k ] will be x[k ] h[n − k ]
50 Copyright © 2005, S. K. Mitra Copyright © 2005, S. K. Mitra TimeDomain Characterization
of LTI DiscreteTime System Convolution Sum • Hence, the response y[n] to an input • The summation ∞ y[n] = k = −∞ will be ∞ k = −∞ which can be alternately written as
51 ∞ k = −∞ y[n] = x[n] * h[n] ∑ x[n − k ] h[k ] k = −∞ ∞ ∑ x[k ] h[n − k ] = ∑ x[n − k ] h[n] is called the convolution sum of the
sequences x[n] and h[n] and represented
compactly as y[n] = ∑ x[k ] h[ n − k ] y[n] = ∞ k = −∞ x[ n] = ∑ x[k ] δ[ n − k ] 52
Copyright © 2005, S. K. Mitra Copyright © 2005, S. K. Mitra Convolution Sum Convolution Sum • Properties • Commutative property:
x[n] * h[n] = h[n] * x[n]
• Associative property :
(x[n] * h[n]) * y[n] = x[n] * (h[n] * y[n])
• Distributive property :
x[n] * (h[n] + y[n]) = x[n] * h[n] + x[n] * y[n]
53 54
Copyright © 2005, S. K. Mitra • Interpretation • 1) Timereverse 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
nth sample of y[n] of the convolution sum
Copyright © 2005, S. K. Mitra 9 Convolution Sum Convolution Sum
• Schematic Representation h[− k ] zn h[n − k ] v[k ]
∑
× y[n] k x[k ] 55 • 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 56 • We illustrate the convolution operation for
the following two sequences:
⎧1, 0 ≤ n ≤ 5
x[ n] = ⎨
⎩0, otherwise
⎧1.8 − 0.3n, 0 ≤ n ≤ 5
h[ n] = ⎨
0,
otherwise
⎩
• Figures on the next several slides the steps
involved in the computation of
y[n] = x[n] * h[n] Copyright © 2005, S. K. Mitra Copyright © 2005, S. K. Mitra Convolution Sum Convolution Sum Plot of x[4 k] and h[k] h[k]x[4 k] 2 3
Amplitude Amplitude 1.5
1
0.5
0
0.5
10 2
1
0 0 10 k→ 10 y[n] 6 6 Amplitude Amplitude 8 4
2
0
10 57 10 k → 0 y[4]
8 0 4
2
0
10 10 0 n 58 10
n Copyright © 2005, S. K. Mitra Copyright © 2005, S. K. Mitra Convolution Sum
3
Amplitude 1.5
Amplitude Plot of x[0 k] and h[k] 1
0.5
0
10 1 k→ 10 3 1.5 2 1
0.5
0 0
0
y[1] h[k]x[0 k] 2
Amplitude 2 0.5
10 Convolution Sum h[k]x[1 k] Amplitude Plot of x[1 k] and h[k] 0 10
y[n] 0.5
10 k→ 2
1
0 0 10
y[0] 10 k→ 10 6 6 6 4
2 0 10
n 4
2
0
10 0 60
Copyright © 2005, S. K. Mitra 4
2
0
10 10
n Amplitude 6 k→ 8 Amplitude 8 Amplitude Amplitude 8 0
10 59 0
y[n] 8 0 10
n 4
2
0
10 0 10
n
Copyright © 2005, S. K. Mitra 10 Convolution Sum
h[k]x[1 k] Amplitude Amplitude 1
0.5
0
0.5
10 Plot of x[3 k] and h[k] 3
2
1 10
y[1] 10 k→ 3 1
0.5
0 0
0 h[k]x[3 k] 2
1.5
Amplitude 2
1.5 Amplitude Plot of x[1 k] and h[k] Convolution Sum 0 10
y[n] 0.5
10 k→ 2
1
0 0 10
y[3] 10 k→ 10 6 6 6 6 4
2
0
10 0 4
2
0
10 10 0 n 61 4
2
0
10 10
n 0 4
2
0
10 10 0 n 62 10
n Copyright © 2005, S. K. Mitra Copyright © 2005, S. K. Mitra Convolution Sum
h[k]x[5 k] 1 0 2
1 10
y[5] 10 k→ 3 1
0.5
0 0
0 h[k]x[7 k] 2
1.5
Amplitude Amplitude Amplitude Plot of x[7 k] and h[k] 3 0.5 0.5
10 Convolution Sum
Amplitude Plot of x[5 k] and h[k]
2
1.5 0 10
y[n] 0.5
10 k→ 2
1
0 0 10 k→ 10 10 6 6 6 6 4
2 0 4
2
0
10 10 0 n 63 4
2
0
10 10
n Amplitude 8 Amplitude 8 0
10 0 4
2
0
10 10 0 n 64 10
n Copyright © 2005, S. K. Mitra Copyright © 2005, S. K. Mitra Convolution Sum
3
Amplitude 1.5
Amplitude Plot of x[10 k] and h[k] 1
0.5
0
10 1 y[9] 10 k→ 3 1.5 2 1
0.5
0 0
0 h[k]x[10 k] 2
Amplitude 2 0.5
10 Convolution Sum h[k]x[9 k] Amplitude Plot of x[9 k] and h[k] 0 10
y[n] 0.5
10 k→ 2
1
0 0
y[10] 10 10 k→ 0 10
y[n] 6 6 6 k→ 8
6 4
2
0
10 0 10
n 4
2
0
10 0 66
Copyright © 2005, S. K. Mitra 4
2
0
10 10
n Amplitude 8
Amplitude 8
Amplitude 8
Amplitude k→ y[n] 8
Amplitude Amplitude 0 y[7] 8 65 k→ 8
Amplitude 8
Amplitude 8
Amplitude Amplitude 0
y[n] 8 0 10
n 4
2
0
10 0 10
n
Copyright © 2005, S. K. Mitra 11 Convolution Sum Convolution Sum
h[k]x[12 k] 1
0.5
0 2
1 0
y[12] 10 10 k→ 3 1
0.5
0 0 0.5
10 h[k]x[13 k] 2
1.5
Amplitude Amplitude Amplitude Plot of x[13 k] and h[k] 3 Amplitude Plot of x[12 k] and h[k]
2
1.5 0 10 0.5
10 k→ y[n] 2
1
0 0
y[13] 10 k→ 10 10 k→ 8 6 6 6 6 4
2
0
10 0 10 4
2
0
10 n 67 0 4
2
0
10 10
n 68 Amplitude 8
Amplitude 8
Amplitude Amplitude 0
y[n] 8 0 4
2
0
10 10 0 n 10
n Copyright © 2005, S. K. Mitra Copyright © 2005, S. K. Mitra TimeDomain Characterization
of LTI DiscreteTime System TimeDomain Characterization
of LTI DiscreteTime System • 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 • 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
timedomain description involving a finite
sum of products will be considered 69 70
Copyright © 2005, S. K. Mitra Copyright © 2005, S. K. Mitra TimeDomain Characterization
of LTI DiscreteTime System TimeDomain Characterization
of LTI DiscreteTime System
• Example  Develop the sequence y[n]
generated by the convolution of the
sequences x[n] and h[n] shown below
x[n] • As can be seen from the shifted timereversed version {h[n − k ]} for n < 0, shown
below for n = −3 , for any value of the
sample index k, the kth sample of either
{x[k]} or {h[n − k ]} is zero h[n] 3 2
3
1 h[−3 − k ] 1 1
0
2 –1 4 n 3
0 1 n 2 2
1 –1 –6 –2 –5 –4 –3 –2 –1 0 k –1 71 72
Copyright © 2005, S. K. Mitra Copyright © 2005, S. K. Mitra 12 TimeDomain Characterization
of LTI DiscreteTime System TimeDomain Characterization
of LTI DiscreteTime System • As a result, for n < 0, the product of the kth
samples of {x[k]} and {h[n − k ]} is always
zero, and hence
y[n] = 0 for n < 0
• Consider now the computation of y[0]
• The sequence
h[ − k ]
2
{h[− k ]} is shown
1
k
on the right
–1 • The product sequence {x[k ]h[−k ]} is plotted
below which has a single nonzero sample
x[0]h[0] for k = 0
x[k ]h[ −k ]
0
–5 –4 –3 –2 –1 73 –2 –1 0 1 2 2 k 3 –2 • Thus y[0] = x[0]h[0] = −2 –3 –6 –5 –4 1 3 74 Copyright © 2005, S. K. Mitra Copyright © 2005, S. K. Mitra TimeDomain Characterization
of LTI DiscreteTime System TimeDomain Characterization
of LTI DiscreteTime System • For the computation of y[1], we shift {h[−k ]}
to the right by one sample period to form
{h[1 − k ]} as shown below on the left
• The product sequence {x[k ]h[1 − k ]} is
shown below on the right • To calculate y[2], we form {h[2 − k ]} as
shown below on the left
• The product sequence {x[k ]h[2 − k ]} is
plotted below on the right –5 –4 –3 –2 –1 1
–2
–1 0 1 2 3 1 2 x[ k ]h[ 2 − k ] 2 0 2 –5 –4 –3 h[ 2 − k ] x[k ]h[1 − k ] h[1 − k ] 3 k 1 1
–1
–4 –3 –2 k 1 0 2 3 4 5 k –3 –2 –1 0 1 2 3 4 5 6 k –1 –1
–4 75 • Hence, y[1] = x[0]h[1] + x[1]h[0] = −4 + 0 = −4 76 y[2] = x[0]h[2] + x[1]h[1] + x[2]h[0] = 0 + 0 + 1 = 1 Copyright © 2005, S. K. Mitra Copyright © 2005, S. K. Mitra TimeDomain Characterization
of LTI DiscreteTime System TimeDomain Characterization
of LTI DiscreteTime System • Continuing the process we get
y[3] = x[0]h[3] + x[1]h[2] + x[2]h[1] + x[3]h[0]
= 2 + 0 + 0 +1 = 3 • From the plot of {h[n − k ]} for n > 7 and the
plot of {x[k]} as shown below, it can be
seen that there is no overlap between these
two sequences
• As a result y[n] = 0 for n > 7 y[4] = x[1]h[3] + x[2]h[2] + x[3]h[1] + x[4]h[0]
= 0 + 0 − 2 + 3 =1
y[5] = x[2]h[3] + x[3]h[2] + x[4]h[1]
= −1 + 0 + 6 = 5
y[6] = x[3]h[3] + x[4]h[2] = 1 + 0 = 1
77 y[7] = x[4]h[3] = −3 x[k] 2 1
0 3
1 2 –1 78
Copyright © 2005, S. K. Mitra h[8 − k ] 3 –2 4 k 1 5 2 3 4 –1 6 7 8 9 10 11 k Copyright © 2005, S. K. Mitra 13 TimeDomain Characterization
of LTI DiscreteTime System TimeDomain Characterization
of LTI DiscreteTime System • Note: The sum of indices of each sample
product inside the convolution sum is equal
to the index of the sample being generated
by the convolution operation
• For example, the computation of y[3] in the
previous example involves the products
x[0]h[3], x[1]h[2], x[2]h[1], and x[3]h[0]
• The sum of indices in each of these
products is equal to 3 • The sequence {y[n]} generated by the
convolution sum is shown below
y[n]
5
3 0 1 1 1 1
7 2 –2 –1 3 4 5 6 8 9 n –2
–3
–4 79 80
Copyright © 2005, S. K. Mitra Copyright © 2005, S. K. Mitra Tabular Method of
Convolution Sum Computation TimeDomain Characterization
of LTI DiscreteTime System • Can be used to convolve two finitelength
sequences
• Consider the convolution of {g[n]}, 0 ≤ n ≤ 3 ,
with {h[n]}, 0 ≤ n ≤ 2 , generating the
sequence y[n] = g[n] * h[n]
• Samples of {g[n]} and {h[n]} are then
multiplied using the conventional
multiplication method without any carry
operation • In the example considered the convolution
of a sequence {x[n]} of length 5 with a
sequence {h[n]} of length 4 resulted in a
sequence {y[n]} of length 8
• 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
81 82
Copyright © 2005, S. K. Mitra Copyright © 2005, S. K. Mitra Tabular Method of
Convolution Sum Computation
n:
g[ n ]:
h[ n ]: y[ n ]: Tabular Method of
Convolution Sum Computation 0
1
2
3
4
5
g[ 0 ]
g[1]
g[ 2 ]
g[ 3 ]
h[ 0 ]
h[1]
h[ 2 ]
g[ 0 ]h[ 0 ] g[1]h[ 0 ] g[ 2 ]h[ 0 ] g[ 3]h[ 0 ]
g[ 0 ]h[1] g[1]h[1] g[ 2 ]h[1] g[ 3 ]h[1]
g[ 0 ]h[ 2 ] g[1]h[ 2 ] g[ 2 ]h[ 2 ] g[ 3]h[ 2 ]
y[ 0 ]
y[1]
y[ 2 ]
y[ 3 ]
y[ 4 ]
y[ 5 ] • The samples of {y[n]} are given by
y[0] = g[0]h[ 0]
y[1] = g[1]h[ 0] + g[ 0]h[1]
y[2] = g[2]h[0] + g[1]h[1] + g[0]h[2]
y[3] = g[3]h[0] + g[2]h[1] + g[1]h[2]
y[ 4] = g[3]h[1] + g[2]h[2]
y[5] = g[3]h[2] • The samples y[n] generated by the
convolution sum are obtained by adding the
entries in the column above each sample
83 84
Copyright © 2005, S. K. Mitra Copyright © 2005, S. K. Mitra 14 Tabular Method of
Convolution Sum Computation Tabular Method of
Convolution Sum Computation • The method can also be applied to convolve
two finitelength twosided sequences
• In this case, a decimal point is first placed
to the right of the sample with the time
index n = 0 for each sequence
• Next, convolution is computed ignoring the
location of the decimal point • Finally, the decimal point is inserted
according to the rules of conventional
multiplication
• The sample immediately to the left of the
decimal point is then located at the time
index n = 0 85 86
Copyright © 2005, S. K. Mitra Copyright © 2005, S. K. Mitra Simple Interconnection
Schemes Convolution Using MATLAB
• The Mfile conv implements the convolution
sum of two finitelength sequences
a =[− 2 0 1 − 1 3
• If • Two simple interconnection schemes are:
• Cascade Connection
• Parallel Connection b =[1 2 0 1]
then conv(a,b) yields
[− 2 − 4 1 3 1 5 1 − 3
87 88
Copyright © 2005, S. K. Mitra Copyright © 2005, S. K. Mitra Cascade Connection Cascade Connection ≡ • 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) h1[n] h2[n] ≡ h2[n] h1[n] h1[n] * h[n]=h2[n]
h[n]
1 • Impulse response h[n] of the cascade of two
LTI discretetime systems with impulse
responses h1[n] and h2[n] is given by
h[n] = h1[n] * h2[n]
89 90
Copyright © 2005, S. K. Mitra Copyright © 2005, S. K. Mitra 15 Cascade Connection Cascade Connection • An application of the inverse system
concept is in the recovery of a signal x[n]
ˆ
from its distorted version x[n] appearing at
the output of a transmission channel
• If the impulse response of the channel is
known, then x[n] can be recovered by
designing an inverse system of the channel • An application is in the development of an
inverse system
• If the cascade connection satisfies the
relation
h1[n] * h2[n] = δ[ n]
then the LTI system h1[n] is said to be the
inverse of h2[n] and viceversa
91 x[ n ]
92 inverse system
channel ^
x[ n ] h1[n] h2[n] h1[n] * h 2[n] = δ[ n] Copyright © 2005, S. K. Mitra Copyright © 2005, S. K. Mitra Cascade Connection Cascade Connection • Example  Consider the discretetime
accumulator with an impulse response µ[n]
• Its inverse system satisfy the condition
µ[n] * h 2[n] = δ[n]
• It follows from the above that h2[n] = 0 for
n < 0 and
h2[0] = 1
n ∑ h2[l] = 0 93 • Thus the impulse response of the inverse
system of the discretetime accumulator is
given by
h2[ n] = δ[n] − δ[n − 1]
which is called a backward difference
system for n ≥ 1 l =0 94
Copyright © 2005, S. K. Mitra Copyright © 2005, S. K. Mitra Simple Interconnection Schemes Parallel Connection
h1[n]
h2[n] 95 x[ n ] + ≡ • Consider the discretetime system where
h1[ n] = δ[n] + 0.5δ[n − 1], h1[n] + h[n]=h2[n]
h[n]
1 • Impulse response h[n] of the parallel
connection of two LTI discretetime
systems with impulse responses h1[n] and
h2[n] is given by
h[n] = h1[n] + h2[n]
Copyright © 2005, S. K. Mitra h2[ n] = 0.5δ[n] − 0.25δ[n − 1],
h3[n] = 2δ[n], h1[n] 96 + h3[n] h4[ n] = −2(0.5) µ[ n]
n + h2[n] h4[n]
Copyright © 2005, S. K. Mitra 16 Simple Interconnection Schemes Simple Interconnection Schemes • Simplifying the blockdiagram we obtain • Overall impulse response h[n] is given by
h[n] = h1[n] + h2[n] * (h3[n] + h4[n])
= h1[n] + h 2[n] * h3[n] + h 2[n] * h 4[n]
• Now, + h1[n] ≡ h2[n] h1[n] + h 2[ n ] * ( h3[ n ]+ h 4[ n ]) h 3[ n ] + h 4[ n ] h2[n] * h3[n] = ( 1 δ[ n] − 1 δ[n − 1]) * 2δ[n]
2 4 = δ[n] − 1 δ[n − 1]
2 97 98
Copyright © 2005, S. K. Mitra Copyright © 2005, S. K. Mitra Simple Interconnection Schemes ( ) h2[n] * h4[n] = ( 1 δ[n] − 1 δ[n − 1]) * − 2( 1 ) n µ[n]
2
2 • Therefore 4
= − ( 1 ) n µ[ n] + 1 ( 1 ) n −1µ[n − 1]
2
22
1 ) n µ[ n] + ( 1 ) n µ[ n − 1]
= − (2
2
= − ( 1 ) n δ[n] = − δ[n]
2 h[ n] = δ[n] + 1 δ[n − 1] + δ[n] − 1 δ[n − 1] − δ[n] = δ[n]
2
2
99
Copyright © 2005, S. K. Mitra 17 ...
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This note was uploaded on 01/11/2010 for the course ECC ECC3107 taught by Professor Drmakhfudzah during the Spring '09 term at Punjab Engineering College.
 Spring '09
 DrMakhfudzah

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