Digital Signal Processing
Discrete Sequences and Systems
Moslem Amiri, Václav Přenosil Masaryk University
Understanding Digital Signal Processing, Third Edition, Richard Lyons (0-13-261480-4) © Pearson Education, 2011.
Discrete Sequences and Their Notation
Signal processing
■ Science of analyzing time-varying physical processes
■ Continuous signal
■ Continuous in time
■ Continuous range of amplitude values
■ Analog (continuous) signal processing
■ Discrete-time signal
■ Time variable is quantized
■ Signal amplitude is quantized
Because we represent all digital quantities with binary numbers, there's a limit to the resolution
■ Digital signal processing
Discrete Sequences and Their Notation
■ Example
■ A continuous sinewave
■ Peak amplitude of 1
■ Frequency f0
x(t) = sm(27rf0t)
m f0 is measured in hertz (Hz) = cycles/second
■ t representing time in seconds
■ f0t has dimensions of cycles
■ 2ixf0t is an angle measured in radians
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Discrete Sequences and Their Notation
Figure 1-1 A time-domain sinewave: (a) continuous waveform representation; (b) discrete sample representation; (c) discrete samples with connecting lines.
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Discrete Sequences and Their Notation
Fig. 1-1
■ Continuous sinewave    sample it once every ts seconds using an analog-to-digital converter
■ Variable f in (a) is continuous
■ Index variable n in (b) is discrete and can have only integer values
■ x(n) is a discrete-time sequence of individual values
■ There is nothing between dots of x(n)
x(t) = sin(2nf0t) —> x(n) = s'm(27rf0nt s)
m x(t) and x(n) are referred to as time-domain signals
Discrete Sequences and Their Notation
Discrete system
■ A collection of hardware components, or software routines, that operate on a discrete-time signal sequence
y(0),y(1),y(2),y(3),...
x(0),x(1),x(2),x(3),
(a)
Discrete System
(b)
x(n)
Discrete System
y(n)
Figure 1-2 With an input applied, a discrete system provides an output: (a) the input and output are sequences of individual values; (b) input and output using the abbreviated notation of x(n) and y(n).
E.g., y(n) = 2x(n)-l
Discrete Sequences and Their Notation
i Given samples of a discrete-time sinewave (e.g., Fig. 1 -1 (b)), find frequency of waveform they represent
■ Possible to say sinewave repeats every 20 samples
■ Not possible to find exact sinewave frequency
We need sample period ts to determine absolute frequency of discrete sinewave
■ If ts = 0.05 milliseconds/sample
. ,   20 samples   0.05 milliseconds sinewave period =-x-= 1 milliseconds
period sample
■ Sinewave's frequency = 1/(1 ms) = 1 kHz
Discrete Sequences and Their Notation
Frequency domain
■ To represent frequency content of discrete time-domain signals
■ Called spectrum
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Discrete Sequences and Their Notation
x1 (n) in the time domain
1 ■■
0.5 ■■ "
(a)   0 -1 l l l | I
5 10 15
-0.5 +
^ x2(n) in the time domain
0.5 4
I— ■ ■ ■
4 tint i ■ ■
(b)   o •mi in
i 5
10 15 a
^-++++4
' " 20
-0.5'
25 30
25
■4-1
AXi(m) amplitude in the frequency domain
0.5 ■■
Time (n)
0    f0  2fQ 3f0 4f0 5f0 Frequency
^ X2(m) amplitude in the
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1 ■■
0.5 ■■
Time (n)
frequency domain
0.4
0    fQ  2f0 3f0 4f0 5f0 Frequency
^ xsum(n) in the time domain 1.5 ■■
1 ■■ .
0.5 ■ ■
20
(C) 0 -0.5 ■ ■ -1 ■■ -1.5 -L
1111' 11' il:-ii+4-
25
<4h
5 10 15
30
4-lH
1 ■■
0.5 ■■
-*sLim(m) amplitude in the
frequency domain
Time (n)     Q ,
0    f0  *o 3/0 4f0 5f0
Frequency
Figure 1-3 Time- and frequency-domain graphical representations: (a) sinewave of frequency fQ; (b) reduced amplitude sinewave of frequency 2fQ; (c) sum of the two sinewaves.
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Discrete Sequences and Their Notation
Fig. 1-3
xsum (n) = xi (n) + x2 (n) = ™(2nf0nts) + 0.4 x sin(2x2 f0nts)
■ xsum(n) nas a frequency component of f0 Hz and a reduced-amplitude frequency component of 2f0 Hz
■ Because x^(n) + x2{n) sinewaves have a phase shift of zero degrees relative to each other, no need to depict this phase relationship in Xsum(A7?)
In general, phase relationships in frequency-domain sequences are important
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Signal Amplitude, Magnitude, Power
■ Amplitude of a variable
■ Measure of how far, and in what direction, that variable differs from zero
■ Can be either positive or negative
■ Magnitude of a variable
■ Measure of how far, regardless of direction, its quantity differs from zero
■ Always positive
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Signal Amplitude, Magnitude, Power
^ \x,(n)\ 1 +-
0.5 --
0
10
15
20
25      30 Time (n)
-0.5 -
Figure 1-4   Magnitude samples, | x^(ri) \, of the time waveform in Figure l-3(a).
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Signal Amplitude, Magnitude, Power
■ In frequency domain, we are often interested in power level of signals
■ Power of a signal is proportional to its amplitude (or magnitude) squared
■ Assuming proportionality constant is one, power of a sequence in time or frequency domains are
V (W) =1 X(W) I2'      Xpwr (m) =1 X(m) I2
■ Often we want to know the difference in power levels of two signals in frequency domain
■ Because of squared nature of power, two signals with moderately different amplitudes will have a much larger difference in their relative powers
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Signal Amplitude, Magnitude, Power
1 --
0.5 ■■
0
Xsum(m) amplitude in the
frequency domain
0.4
1 ■-
0.5 ■■
0
Xsum(m) power in the
frequency domain
0.16
0     'o  2fo 3fo 4'o 5fo
Frequency
-I-■—■—"-^-►
0    fo 2fo 3fo 4fo 5fo
Frequency
Figure 1-5   Frequency-domain amplitude and frequency-domain power of the xsum(n) time waveform in Figure l-3(c).
■ Because of their squared nature, plots of power values often involve showing both very large and very small values on same graph
■ To make these plots easier to generate and evaluate, decibel scale is usually employed
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Signal Processing Operational Symbols
■ Block diagrams
■ Are used to graphically depict the way digital signal processing operations are implemented
■ Comprise an assortment of fundamental processing symbols
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Signal Processing Operational Symbols
(a)
b(n)
Addition:
-+\ + J-► a(n) a{n) = b{n) + c(n)
c(n)
(b)
Subtraction:
-T
c(n)
a(n)      a(n) = b(n) - c(n)
(c)
n+3 a(n) =}J)(k) k= n
= b(n) + b(n+1) + b(n+2) + fo(n+3)
bin)
(d)
Multiplication: -► a(n)
c(n)
a(n) = b(n)c{n) = b{n)- c(ri) [Sometimes we use a " ■" to signify multiplication.]
(e)
b{ri)
b{n)
Unit delay:
Delay
a(n) a(n)
a(n) = to(/>1)
Figure 1-6   Terminology and symbols used in digital signal processing block diagrams.
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Discrete Linear Time-Invariant Systems
■ Linear time-invariant (LTI) systems
■ Vast majority of discrete systems used in practice are LTI systems
■ LTI systems are very accommodating when it comes to their analysis
■ We can use straightforward methods to predict performance of any digital signal processing scheme as long as it's linear and time invariant
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Discrete Linear Systems
Linear
■ A linear system's output resulting from a sum of individual inputs is superposition (sum) of individual outputs
z \     results in   v     / \
->yM)
z \     results in   v      / \
x2(n)->y2(n)
/ \ i      / \     results in   v     / \ ,      / \
xx (n) + x2 (n)-> yx (n) + y2(n)
m Also, if inputs are scaled by constant factors c1 and c2, output sequence parts are scaled by those factors too
/ \ , / \     results in   v / \ , / \
cxxx {n) + c2x2 (n)-> c2yx (n) + c2y2 (n)
1 O
Discrete Linear Systems
(a)
Input x(n)
Linear	
Discrete	
System	
Output y(n) = -x(n)/2
Figure 1-7 Linear system input-to-output relationships: (a) system block diagram where y(ri) = -x(n)/2; (b) system input and output with a 1 Hz sinewave applied; (c) with a 3 Hz sinewave applied; (d) with the sum of 1 Hz and 3 Hz sinewaves applied. 19
Discrete Linear Systems
■ Linearity in Fig. 1-7(d)
■ x3(n) input sequence is sum of a 1 Hz sinewave and a 3 Hz sinewave
■ Thus y3(n) is sample-for-sample sum of y^(n) and
y2(n)
m Also output spectrum Y3(at7) is sum of Y^m) and Y2(m)
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Discrete Linear Systems
(a)
Input x(n)
Nonlinear Discrete System
Output y(n) = [x(n)]"
Figure 1-8 Nonlinear system input-to-output relationships: (a) system block diagram where y(n) = (x(n))2; (b) system input and output with a 1 Hz sinewave applied; (c) with a 3 Hz sinewave applied; (d) with the sum of 1 Hz and 3 Hz sinewaves applied, 21
Discrete Linear Systems
y\ O) =
Fig. 1-8(b)
xx (n) = sin(27f0nts) = sin(2^ x 1 x nts)
yx (n) = [xj (/?)]2 = sin(2^- x 1 x nts) x sin(2^- x 1 x nts)
. , x . cos(a-6) cos(a + B) sm(a) x sin(p) =---
2 2
cos(2;z" x 1 x «^ - 2n x 1 x «^ )   cos(2^ x 1 x nts + 2^ x 1 x nts)
2 2 cos(O)   cos(4^xlx^)    1 cos(2^x2x^)
2 2 ~ 2 2
■ y^n) is a cosine wave of 2 Hz and a peak amplitude of -0.5, added to a constant value (zero Hz) of 1/2
Fig. 1-8(c)
■ Y2(n) contains a zero Hz and a 6 Hz component 22
Discrete Linear Systems
Fig. 1-8(d)
■ x3(n) comprises sum of a 1 Hz and a 3 Hz sinewave
a = \Hz sinewave, b = 3 Hz sinewave —> (a + b) = a + lab + b
a —> zero Hz and 2 Hz
b —» zero Hz and 6 i/z
2aZ? = 2 sin(2;z" x 1 x nts) x sin(2^ x 3 x nts)
2 cos(2;r x 1 x «^ - 2n x 3 x nts)   2 cos(2^ x 1 x «^ + 2^ x 3 x nts)
2 2 = cos(2;r x2xnts)- cos(2^ x4x«/J
2aZ? -^2 Hz and 4 #z
■ Two additional sinusoids are present in y3(n) because of system's nonlinearity, a 2 Hz cosine wave (amp=+1), a 4 Hz cosine wave (amp=-1)
Time-Invariant Systems
Time-invariant system
■ A time delay (or shift) in input sequence causes an equivalent time delay in system's output sequence
/ \     results in   v    / \
x(n)->y(n)
x (n) = x(n + k)—results in > y (w) = y(n + k)
■ k is some integer representing k sample period time delays
■ For a system to be time invariant, above equation must hold true for any integer value of k and any input sequence
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Time-Invariant Systems
(a)
Input x(n)
Linear Time-Invariant Discrete System
Output y(n) = -x{n)/2
kx(n)
(b) 0 -1
4    6    8 10
0 2
12 14 Time
0.5 0
-0.5
iy(n)
12 14.
4    6    8 10
Time
A x'(n)
(c) 0 -1
■ . 8   10   12 14
0    2    4 6
Time
0.5*
0
0 2 -0.5 i
4 6
8   10   12 14 Time
Figure 1-9 Time-invariant system input/output relationships: (a) system block diagram, y(ri) = -x(n)/2; (b) system input/output with a sinewave input; (c) input/output when a sinewave, delayed by four samples, is the input.
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Time-Invariant Systems
Fig. 1-9
■ Input sequence x'{n) is equal to sequence x{n) shifted to right by k = -4 samples
x (n) = x(n-4)
m System is time invariant because y"{n) output sequence is equal to y{n) sequence shifted to right by four samples
y\n) = y(n-4)
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Commutative Property of LTI Systems
LTI systems have a useful commutative property
■ Their sequential order can be rearranged with no change in their final output
Input x(n)	LTI	An)	LTI
	System #1		System #2
	Input x(n)	LTI	Sin)	LTI	Output y(n)
(b)		System #2		System #1	
Figure 1-10   Linear time-invariant (LTI) systems in series: (a) block diagram of two
LTI systems; (b) swapping the order of the two systems does not change the resultant output y(ri).
Analyzing LTI Systems
■ Unit impulse response of an LTI system
■ System's time-domain output sequence when input is a single unity-valued sample (unit impulse) preceded and followed by zero-valued samples
■ System's unit impulse response completely characterizes the system
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Analyzing LTI Systems
(a)
Input x(n)
Linear	
Time-Invariant	
Discrete	
System	
Output y(n)
(b)
iix(n) impulse input 1-
^unity-valued sample
Ay(n) impulse response
0
Time
0
Time
Figure 1-11 LTI system unit impulse response sequences: (a) system block diagram; (b) impulse input sequence x(n) and impulse response output sequence y(n).
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Analyzing LTI Systems
■ Knowing impulse response, we can determine system's output for any input
■ Output is equal to convolution of input sequence and system's impulse response
■ Moreover, we can find system's frequency response by taking discrete Fourier transform of that impulse response
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Analyzing LTI Systems
x(n-1)
x(n-2)
x(n-3)
(a)
A x(n) impulse input
1
(b)
(c)
iiy(n) impulse response
1/4
unity-valued sample
A \Y(m)\
1
0.75+ 0.5 0.25 0
n (Time) 0 n (Time)
/
Discrete Fourier transform
Frequency magnitude response
—
0
H—h
3
H—I—I—I—I—I—h
6      9      12     15     18 m(Freq)
Figure 1-12 Analyzing a moving averager: (a) averager block diagram; (b) impulse input and impulse response; (c) averager frequency magnitude response.
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Analyzing LTI Systems
Fig. 1-12
■ A4-point moving averager
1 1 v,
y(n) = — [x(n) + x(n -1) + x(n — 2) + x(n - 3)] = — > x(k)
k=n-3
m Frequency magnitude response plot shows that moving averager has characteristic of a lowpass filter
■ Averager attenuates (reduces amplitude of) high-frequency signal content applied to its input
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