Bài giảng Digital Signal Processing - Chapter 3: Discrete. Time Systems - Võ Trung Dũng

1Digital Signal Processing
Discrete-Time Systems
Dr. Dung Trung Vo
Telecommunication Divisions
Department of Electrical and Electronics
September, 2013
Input/Output Rules
 Discrete-time system: is a processor that transforms an input sequence of 
discrete-time samples x(n) into an output sequence of samples y(n):
 Sample-by-sample processing: 
 Block processing methods:
 Functional mapping: 
 Linear systems: this mapping becomes a linear transformation
   ,...,...,,,,,...,...,,,, 32103210 nHn yyyyyxxxxx 
 xHy 
Hxy 
 Example 1: y(n)= 2x(n), sample-by-sample processing
 Example 2: y(n)= 2x(n)+3x(n − 1)+4x(n − 2), block processing
 Example 3: y(n)= x(2n)
Input/Output Rules
   ,...,2,2,2,2,...,,,, 32103210 xxxxxxxx H
   ,...,,,,...,,,,,, 64206543210 xxxxxxxxxxx H
 Linear system: has the property that the output signal due to a linear 
combination of two or more input signals can be obtained by forming the same 
linear combination of the individual outputs. if y1(n) and y2(n) are the outputs due 
to the inputs x1(n) and x2(n), then the output due to the linear combination of 
inputs
is given by the linear combination of outputs
 Testing linearity:
Linearity and Time Invariance
)()()( 2211 nxanxanx 
)()()( 2211 nyanyany 
2 Example 3.2.1: Test the linearity of the discrete-time systems defined by 
and
 Solution: the output due to the linear combination will be
is not equal to the linear combination of each input
So it is not a linear system.
Similarly, for the quadratic system
Linearity and Time Invariance
  3)()(23)(2)( 2211  nxanxanxny
)3)(2()3)(2()()( 22112211  nxanxanyanya
3)(2)(  nxny )()( 2 nxny 
)()( 2 nxny 
 22211222211 )()()()( nxanxanxanxa 
 Time Invariance system: is a system that remains unchanged over time. This 
implies that if an input is applied to the system today causing a certain output to be 
produced, then the same output will also be produced tomorrow if the same input is 
applied
 Right translation:
Testing time invariance: delaying the input causes the output to be delayed by 
the same amount:
Linearity and Time Invariance
 Example 3.2.2: Test the time invariance of the discrete-time systems defined by 
and
 Solution:
 Delaying the input signal
 Delaying the output signal
Thus, the system is not time-invariant.
Similarly, for system y(n)= x(2n).
Thus, the downsampler is not time-invariant
Linearity and Time Invariance
)()()( Dnnxnnxny DD 
)()()()()( nyDnnxDnxDnDny D
)2()2()( Dnnxnnxny DD 
)()2())(2()( nyDnxDnxDny D
)()( nnxny  )2()( nxny 
 Impulse response of an LTI system: Linear time-invariant systems are 
characterized uniquely by their impulse response sequence h(n), which is defined as 
the response of the system to a unit impulse δ(n).
 Transform function:
 Sample-by-sample processing:
Impulse Response
)()( nhn H




0,0
0,1
)(
nif
nif
n
   ,...,,,,...0,0,0,1 3210 hhhhH
3 Delayed impulse responses: time invariance implies
Impulse Response
)()( DnhDn H 
 Impulse response of an LTI system: linearity implies
 Arbitrary input: {x(0), x(1), x(2), . . . } can be thought of as the linear 
combination of shifted and weighted unit impulses:
 Its output:
Impulse Response
)2()1()()2()1()(  nhnhnhnnn H
...)2()2()1()1()()0()(  nhxnhxnhxny
...)2()2()1()1()()0()(  nxnxnxnx 
LTI form of convolution:
Summation could extend over negative values of m, depending on the input signal.
 Direct form of convolution:
Impulse Response
 
m
mnhmxny )()()(
 
m
mnxmhny )()()(
 FIR: has impulse response h(n) that extends only over a finite time interval, say 0 
≤ n ≤ M, and is identically zero beyond that
 Filter order: M
 Filter length: length of the impulse response vector h = [h0 , h1, . . . , hM]
 Impulse response coefficients: h = [h0 , h1, . . . , hM] are referred to by 
various names, such as filter coefficients, filter weights, or filter taps
 FIR filtering equation:
FIR and IIR Filters



M
m
mnxmhny
0
)()()(
1 MLh
 ,...0,0,0,,...,,,, 3210 Mhhhhh
4 IIR: has an impulse response h(n) of infinite duration, defined over the infinite 
interval 0 ≤ n < ∞
 IIR filtering equation:
 Constant coefficient linear difference equations: This I/O equation is not 
computationally feasible. Therefore, we must restrict our attention to a subclass of 
IIR filters, namely, those filter coefficients are coupled to each other through 
constant coefficient linear difference equations:
 Difference equation for y(n):
FIR and IIR Filters



0
)()()(
m
mnxmhny



L
i
i
M
i
i inbinhanh
11
)()()( 



L
i
i
M
i
i inxbinyany
11
)()()(
 Example 1: Determine the impulse response h of the following FIR filters
Solution: impulse response coefficients
Unit impulse as input, x(n)= δ(n):
FIR and IIR Filters
)4()()()
)3(2)2(5)1(3)(2)()


nxnxnyb
nxnxnxnxnya
   
   1,0,0,0,1,,,,)
2,5,3,2,,,)
43210
3210


hhhhhhb
hhhhha
)4()()()
)3(2)2(5)1(3)(2)()


nnnhb
nnnnnha


 Example 2: Determine the I/O difference equation of an IIR filter whose impulse 
response coefficients h(n) are coupled to each other by the difference equation
 Solution: Setting n = 0, we have h(0)= h(−1)+δ(0)= h(−1)+1. Assuming causal 
initial conditions, h(−1)= 0, we find h(0)= 1. For n > 0, the delta function vanishes, 
δ(n)= 0, and therefore, the difference equation reads h(n)= h(n−1)
 Convolutional I/O equation:
 Recursive difference equation
FIR and IIR Filters
 




00
)()()()(
mm
mnxmnxmhny
)()1()(
...)3()2()1()()(
nxnyny
nxnxnxnxny


)()1()( nnhnh 




0,0
0,1
)()(
nif
nif
nunh
 Example 3: Suppose the filter coefficients h(n) satisfy the difference equation
where a is a constant. Determine the I/O difference equation relating a general 
input signal x(n) to the corresponding output y(n).
Solution: filter coefficients
I/O difference equation
FIR and IIR Filters
)()1()( nnahnh 
 
)()1()(
...)3()2()1()()(
...)3()2()1()()(
2
32
nxnayny
nxanaxnxanxny
nxanxanaxnxny







0,0
0,
)()(
nif
nifa
nuanh
n
n
5 Causal signal: A causal or right-sided signal x(n) exists only for n ≥ 0 and 
vanishes for all negative times n ≤ −1
 Anti-causal signal: An anticausal or left-sided signal exists only for n ≤ −1 and 
vanishes for all n ≥ 0.
 Mixed signal: A mixed or double-sided signal has both a left-sided and a right-
sided part
LTI systems can also be classified in terms of their causality properties depending 
on whether their impulse response h(n) is causal, anticausal, or mixed
Causality
 Description: a system is stable if the output remains bounded by some bound 
|y(n)| ≤ B if its input is bounded, say |x(n)| ≤ A. 
 Necessary and sufficient condition:
Stability is absolutely essential in hardware or software implementations of LTI 
systems because it guarantees that the numerical operations required for computing 
the I/O convolution sums or the equivalent difference equations remain well 
behaved and never grow beyond bounds.
The concepts of stability and causality are logically independent, but are not always 
compatible with each other
Stability

n
nh )(
 Example 4: Consider the following four examples of h(n):
 Solution:
Stability
 
 




























5.01
5.05.02)(
2)(
25.0)(
5.01
15.0)(
11
0
11
m
n
n
n
n
n
n
n
m
m
n
n
n
n
n
n
nh
nh
nh
nh
 Homework: 3.1, 3.2, 3.3, 3.4, 3.6
Stability