EcE 5013 Digital Signal Processing

1
EcE 5013 Digital Signal Processing

• Daw Mya Mya Aye
• Deputy Professor
• Department of Electronic Engineering
  Information Technology
• Yangon Technological University

2
Overview

• Continuous or analog signals
• Discrete-time signals
• Basic analog signals
• Basic discrete signals
• Quantized signals
• Digital signals
• Causal signals
• Random signals
• Representation of signals in MATLAB
• Definition of a system
• Block diagram

3
What is a signal?

• A signal is a physical quantity, or quality,
  which conveys information
• Example voice of my friend is a signal which
  causes me to perform certain actions or react in
  a particular way
• My friend's voice is called an excitation
• My action or reaction is called a response

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What is signal processing?

• The conversion from excitation to response is
  called signal processing
• A typical reason for signal processing is to
  eliminate or reduce an undesirable signal
• We convert the original signal into a form that
  is suitable for further processing
• One fundamental representation of a signal is as
  a function of at least one independent variable

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What is a waveform

• The variation of the signal value as a function
  of the independent variable is called a waveform
• The independent variable often represents time
• We define a signal as a function of one
  independent variable that contains information
  about the behavior or nature of a phenomenon
• We assume that the independent variable is time
  even in cases where the independent variable is a
  physical quantity other than time

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Continuous or analog signals

• Continuous signal is a signal that exists at
  every instant of time
• A continuous signal is often referred to as
  continuous time or analog
• The independent variable is a continuous
  variable
• Continuous signal can assume any value over a
  continuous range of numbers

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Discrete-time signals

• A signal defined only for discrete values of time
  is called a discrete-time signal or simply a
  discrete signal
• Discrete signal can be obtained by taking samples
  of an analog signal at discrete instants of time
• Digital signal is a discrete-time signal whose
  values are represented by digits

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What is sampling?

• Sampling is capturing a signal at an instant in
  time
• Sampling means taking amplitude values of the
  signal at certain time instances
• Uniform sampling is sampling every T units of
  time

Sampling frequency or sampling rate
time step or sample interval
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Sinusoidal signal
Phase in radian (rad)
Amplitude
Time in seconds (s)
Frequency in Hertz (Hz)
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MATLAB code for sine signal

• Xs 1.8
• fs 10
• fi pi/3
• t1 -0.1
• tstep 0.01
• t2 0.2
• t t1tstept2
• x Xssin(2pifstfi)
• plot(t, x)
• xlabel('t')
• ylabel('x_s')
• title('x_s(t) X_s sin(2 \pi f_s t \phi_s)')
• grid on

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Advanced MATLAB code

• Xs 1.8
• fs 10
• fi pi/3
• t1 -0.1
• t2 0.2
• t t1, t2
• x inline('Xssin(2pifstfi)','t','Xs','fs','f
  i')
• fplot(x,t,2e-3,1,'-',Xs,fs,fi)
• xlabel('t') ylabel('x_s') grid on
• title('x_s(t) X_s sin(2 \pi f_s t \phi_s)')

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Exponential signal
x inline('Xeexp(bt)','t','Xe','b') Xe
0.8 b -0.5 t1 0 t2 8 t t1,
t2 fplot(x,t,2e-3,1,'-',Xe,b) xlabel('t') ylabe
l('x_e') title('x_e(t) X_e eb t') grid on
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Unit step signal
x inline('tgt0', 't') t1 -2 t2 6 t
t1, t2 fplot(x, t) xlabel('t') ylabel('u')
title('Unit step signal') axis(t -0.1 1.1)
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Pulse signal
x inline('(1/e)((tgt0)(tlte))','t','e') e
1/100 t1 -1 t2 5 t t1,
t2 fplot(x,t,1e-5,1000,'-',e)
set(gca,'FontSize',16) xlabel('t') ylabel('p_\eps
ilon(t)') axis(t -0.1 1.1/e) title('Pulse
function, \epsilon 1/100')
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Unit impulse signal (Dirac delta)
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Causal signals

• A signal is causal if it is zero for t lt 0
• Causal signals are readily created by multiplying
  any continuous signal by the unit step signal
• The instant when the signal begins is called the
  starting time
• We usually take the starting time to be zero

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Causal signals in MATLAB
B 0.02 a 0.1 f 0.53 phi 3pi/4 t
-50.0510 x Baexp(-at).sin(...
2piftphi) xu x.(tgt0) u
(tgt0) plot(t,x,t,u,t,xu) ylabel('x(t)') xlabel('t
(s)') text(0,1.2,'u(t)') text(-4,-1.1,'x(t)') tex
t(5,-.6,'x(t)u(t)') axis(t(1) t(end) -1.5 1.5)
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Discrete-time signal Sequence

• A sequence (discrete-time signal, discrete
  signal, data sequence, or sample set) is a
  collection of ordered samples
• In practical applications we process
  finite-length sequences
• The existing sequence is often a sampled version
  of a continuous signal

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Sinusoidal sequence
Phase in radian (rad)
Amplitude
Sample index
Period
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Exponential sequence
Xe 0.8 a 0.75 k1 0 k2 10 k
k1k2 x Xea.k stem(k,
x) xlabel('k') ylabel('x_e') title('x_e,k X_e
ak')
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Unit step sequence
k1 -5 k2 10 k k1k2 x (kgt0)
stem(k, x) xlabel('k') ylabel('u_k') title('Uni
t step sequence') axis(k1 k2 -0.1 1.1)
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Unit impulse sequence
k1 -5 k2 10 k k1k2 x (k0)
stem(k, x) xlabel('k') ylabel('\delta_k') title
('Unit impulse sequence') axis(k1 k2 -0.1 1.1)
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Causal sequence

• A sequence that is nonzero only over a finite
  interval of indices is called a finite-length
  sequence
• A sequence whose samples are zero-valued for
  negative indices is causal
• Anti-causal sequence can have nonzero samples
  only for negative indices

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Causal sequences in MATLAB
B 0.94 a 0.1 f 0.32 k -10120 x
(B.k).cos(2pifk) u (kgt0) ux
x.u subplot(3,1,1) stem(k,u,'g') ylabel('u_k','F
ontSize',14) axis(k(1) k(end) -2
2) subplot(3,1,2) stem(k,x,'b') ylabel('x_k','Fon
tSize',14) subplot(3,1,3) stem(k,ux,'r') ylabel('x
_k u_k','FontSize',14) xlabel('k') axis(k(1)
k(end) -2 2)
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Quantized signal

• The purpose of sampling a continuous signal is to
  transmit, store, or process a limited number of
  samples that are represented by a limited number
  of digits
• By using fewer digits we attain faster
  transmission and smaller storage requirements for
  the information
• We utilize the quantized samples rather than the
  true samples of infinite accuracy

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Choice of digits for quantization

• Choice of digits for quantization should be done
  properly in transmitting, storing, and
  processing we prefer less digits
• With too small a number of digits we can lose
  information from the original signal
• Two opposing requirements must be satisfied
• Minimize number of digits to facilitate the
  signal transmission or storing, and
• Maximize number of digits to keep the
  quantization error as low as necessary in order
  to preserve the information

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Digital signal in MATLAB
t 030 x 0.22sin(0.245t0.15) d
0.5 xq dround(x/d) plot(t,x) hold
on stem(t,xq,'r') hold off ylabel('x(t),
x_q(kT)') xlabel('t') legend('analog signal',...
'digital (quantized)')
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Deterministic and random signal

• Signal that can be described by an explicit
  mathematical form is deterministic
• Deterministic signal can be periodic or aperiodic
• Periodic signal consists of a basic shape of
  finite duration that is replicated infinitely
• Signal that cannot be described in an explicit
  mathematical form is called random, also known as
  nondeterministic or stochastic

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Random signal in MATLAB
xk 0150 x rand(size(k)) m mean(x) s
std(x) stem(k,x) hold on plot(k(1) k(end),
m m,'r',... k(1) k(end), s
s,'g') hold off xlabel('k') ylabel('x_k') ytick
0 s m 1 set(gca,'YTick',ytick) legend('random
seq', 'mean','std') title('Uniformly
distributed samples')
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Random signal in MATLAB
k 0150 x randn(size(k)) m mean(x) s
std(x) stem(k,x) hold on plot(k(1) k(end),
m m,'r', k(1) k(end), s
s,'g') hold off xlabel('k') ylabel('x_k') legend(
'random seq', 'mean','std') ytick
sort(-2 s m 2) set(gca,'YTick',ytick) title('No
rmally distributed samples')
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What is a system?

• A signal is a physical quantity, or quality,
  which conveys information
• Systems take one or more signals as input,
  perform operations on the signals, and produce
  one or more signals as output
• A system is a group of related parts working
  together, or an ordered set of ideas, methods, or
  ways of working

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Definition of a system

• Implementation point-of-view a system is an
  arrangement of physical components connected or
  related in such a manner as to form and/or act as
  an entire unit
• Signal processing perspective a system can be
  viewed as any process that results in the
  transformation of signals, in which systems act
  on signals in prescribed ways
• Mathematical a system as a mapping of N input
  signals onto M output signals the mapping
  carries out a transformation on the input signals
  according to a set of rules

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Basic definitions

• Single-variable system (SISO system) has only
  one input and only one output
• Multivariable system (MIMO system) has more than
  one input or more than one output
• Input-output relationship (external description)
  is an equation that describes the relation
  between the input and the output of a system
• Black box concept the knowledge of the internal
  structure of a system is unavailable the only
  access to the system is by means of the input
  ports and the output ports

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Time response

• One-dimensional system required for processing a
  signal that is a function of the single
  independent variable
• We assume that the independent variable is time
  even in cases where the independent variable is a
  physical quantity other than time
• Time response is the output signal as a function
  of time, following the application of a set of
  prescribed input signals, under specified
  operating conditions

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Continuous-time system
Continuous-time system the input and output
signals are continuous time
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Discrete-time system
Discrete-time system has discrete-time input and
output signals
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Digital system

• A discrete-time system is digital if it operates
  on discrete-time signals whose amplitudes are
  quantized
• Quantization maps each continuous amplitude
  level into a number
• The digital system employs digital hardware
• explicitly in the form of logic circuits
• implicitly when the operations on the signals are
  executed by writing a computer program

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Analysis and design

• Analysis of a system is investigation of the
  properties and the behavior (response) of an
  existing system
• Design of a system is the choice and arrangement
  of systems components to perform a specific task
• Design by analysis is accomplished by modifying
  the characteristics of an existing system
• Design by synthesis we define the form of the
  system directly from its specifications

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Block diagram

• Block diagram is a pictorial representation of a
  system that provides a method for characterizing
  the relationships among the components
• Single block with one input and one output is the
  simplest form of the block diagram
• Interior of the rectangle representing the block
  contains (a) component name, (b) component
  description, or (c) the symbol for the
  mathematical operation to be performed on input
  to yield output
• Arrows represent the direction of signal flow

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Elements of block diagram
Takeoff point
Summing point
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Interconnections of blocks
Blocks connected in cascade
Blocks connected in feedback
Blocks connected in parallel
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State

• For some systems, the output at time t0 depends
  not only on the input applied at t0, but also on
  the input applied before t0
• The state is the information at t0 that, together
  with input for t t0, determines uniquely output
  for t t0
• Dynamical equation is the set of equations that
  describes unique relations between the input,
  output, and state

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Relaxed system

• A system is said to be relaxed at time t0 if the
  output for t t0 is solely and uniquely
  determined by the input for t t0
• If the concept of energy is applicable, the
  system is said to be relaxed at t0 if no energy
  is stored in the system at t0
• A system is said to be zero-input if the output
  for t t0 is solely and uniquely determined by
  the state

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Causality and stability

• A system is called causal if the output depends
  only on the present and past values of the input
• Intuitively, a stable system is one that will
  remain at rest unless excited by an external
  source and will return to rest if all excitations
  are removed
• A relaxed system is BIBO stable (bounded-input
  bounded-output) if every bounded input produces a
  bounded output

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Time-invariant system

• A relaxed system is time-invariant if a time
  shift in the input signal causes a time shift in
  the output signal
• In the case of discrete-time digital systems, we
  often use the term shift-invariant instead of
  time-invariant
• Characteristics and parameters of a
  time-invariant system do not change with time

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Linear system

• Consider a relaxed system in which there is one
  independent variable t
• A linear system is a system which has the
  property that if
• input x1(t) produces an output y1(t) and
• input x2(t) produces an output y2(t), then
• input c1 x1(t) c2 x2(t) produces an output c1
  y1(t) c2 y2(t) for any x1(t), x2(t) and
  arbitrary constants c1 and c2

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Principle of superposition

• The response y(t) of a linear system due to
  several inputs x1(t), x2(t), xN(t) acting
  simultaneously is equal to the sum of the
  responses of each input acting alone
• If yi(t) is the response due to the input xi(t),
  then

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Linear time-invariant (LTI) system

• Continuous-time system is LTI if its input-output
  relationship can be described by the ordinary
  linear constant coefficient differential equation

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continued

• Discrete-time system is LTI if its input-output
  relationship can be described by the linear
  constant coefficients difference equation

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Response of an LTI system

• Free response (zero-input response) is the
  solution of the differential equation when the
  input is zero
• Forced response (zero-state response) is the
  solution of the differential equation when the
  state is zero
• Total response is the sum of the free response
  and the forced response
• Total response can be viewed, also, as the sum of
  the steady-state response and transient response
• Steady-state response is that part of the total
  response which does not approach zero as time
  approaches infinity
• Transient response is that part of the total
  response which approaches zero as time goes to
  infinity

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Procedure for analyzing a system

• Determine the equations for each system
  component
• Choose a model for representing the system (e.g.,
  block diagram)
• Formulate the system model by appropriately
  connected the components
• Determine the system characteristics

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Convolution integral

• Unit impulse response is the output of a
  continuous-time LTI system to a unit impulse
  input when the state is zero
• If we know the input and impulse response of a
  causal LTI system, the forced response can be
  found by the convolution integral

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Transient system specifications

• Unit step response is the output of an LTI system
  to a unit step input when the state is zero
• Overshoot, Delay time, Rise time, Settling time

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Continued
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Continued
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Convolution sum

• Unit impulse response is the output of a
  discrete-time LTI system to a unit impulse input
  when the state is zero
• If we know the input and impulse response of a
  causal LTI system, the forced response can be
  found by the convolution sum

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Continued
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Further reading

• M. D. Lutovac, D. V. Tošic, B. L. Evans
• Filter Design for Signal Processing Using MATLAB
  and Mathematica
• Prentice HallUpper Saddle River, New Jersey
  ISBN 0-201-36130-2, (c) 2001