Digital Signal Processing
Moslem Amiri, Václav Přenosil
Masaryk University
Understanding Digital Signal Processing, Third Edition, Richard Lyons
(0-13-261480-4) © Pearson Education, 2011.
Discrete Sequences and Systems
2
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
3
Discrete Sequences and Their Notation
 Example
 A continuous sinewave
 Peak amplitude of 1
 Frequency fo
 fo is measured in hertz (Hz) = cycles/second
 t representing time in seconds
 fot has dimensions of cycles
 2πfot is an angle measured in radians
)2sin()( tftx o
4
Discrete Sequences and Their Notation
continuous sinewave
 sample it once
every ts seconds
using an analog-todigital
converter
variable t is continuous
Index variable n 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)
)2sin()(
)2sin()(
so
o
ntfnx
tftx




x(t) and x(n) are
referred to as timedomain
signals
5
Discrete Sequences and Their Notation
 Discrete system
 A collection of hardware components, or software
routines, that operate on a discrete-time signal
sequence
 E.g., 1)(2)(  nxny
6
Discrete Sequences and Their Notation
 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
 Sinewave’s frequency = 1/(1 ms) = 1 kHz
dsmillisecon1
sample
dsmillisecon0.05
period
samples20
periodsinewave 
7
Discrete Sequences and Their Notation
 Frequency domain
 To represent frequency content of discrete timedomain
signals
 Called spectrum
8
Discrete Sequences and Their Notation
xsum(n) has a frequency
component of fo Hz and a
reduced-amplitude
frequency component of
2fo Hz
)22sin(4.0)2sin()()()( 21 sososum ntfntfnxnxnx  
Because x1(n) + x2(n)
sinewaves have a phase
shift of zero degrees
relative to each other, no
need to depict this phase
relationship in Xsum(m)
(In general, phase
relationships in
frequency-domain
sequences are important)
9
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
10
Signal Amplitude, Magnitude, Power
11
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
 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
22
|)(|)(,|)(|)( mXmXnxnx pwrpwr 
12
Signal Amplitude, Magnitude, Power
 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
13
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
14
Signal Processing Operational Symbols
15
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
16
Discrete Linear Systems
 Linear
 A linear system’s output resulting from a sum of
individual inputs is superposition (sum) of
individual outputs
 Also, if inputs are scaled by constant factors c1
and c2, output sequence parts are scaled by
those factors too
)()()()(
)()(
)()(
2121
22
11
nynynxnx
nynx
nynx
inresults
inresults
inresults
 
 
 
)()()()( 22122211 nycnycnxcnxc inresults
 
17
Discrete Linear Systems
linearity:
x3(n) input sequence is sum of
a 1 Hz sinewave and a 3 Hz
sinewave
thus y3(n) is sample-forsample
sum of y1(n) and y2(n)
also output spectrum Y3(m)
is sum of Y1(m) and Y2(m)
18
Discrete Linear Systems
19
Discrete Linear Systems
 Fig. 1-8(b)
 y1(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
2
)22cos(
2
1
2
)14cos(
2
)0cos(
2
)1212cos(
2
)1212cos(
)(
2
)cos(
2
)cos(
)sin()sin(
)12sin()12sin()]([)(
)12sin()2sin()(
1
2
11
1
ss
ssss
ss
sso
ntnt
ntntntnt
ny
ntntnxny
ntntfnx




















20
Discrete Linear Systems
 Fig. 1-8(d)
 x3(n) comprises sum of a 1 Hz and a 3 Hz
sinewave
 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)
HzHzab
ntnt
ntntntnt
ntntab
HzHzzerob
HzHzzeroa
bababaHzbHza
ss
ssss
ss
4and22
)42cos()22cos(
2
)3212cos(2
2
)3212cos(2
)32sin()12sin(22
6and
2and
2)(sinewave3,sinewave1
2
2
222













21
Time-Invariant Systems
 Time-invariant system
 A time delay (or shift) in input sequence causes
an equivalent time delay in system’s output
sequence
 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
)()()()(
)()(
''
knynyknxnx
nynx
inresults
inresults
 
 
22
Time-Invariant Systems
input sequence x′(n) is equal
to sequence x(n) shifted to
right by k = −4 samples
x′(n) = x(n − 4)
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)
23
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
24
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
25
Analyzing LTI Systems
26
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
27
Analyzing LTI Systems
a 4-point moving averager


n
nk
kxnxnxnxnxny
3
)(
4
1
)]3()2()1()([
4
1
)(
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