Chapter 2. Discrete-Time Signals and Systems

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Chapter 2. Discrete-Time Signals and Systems

• Gao Xinbo
• School of E.E., Xidian Univ.
• Xbgao_at_ieee.org
• http//see.xidian.edu.cn/teach/matlabdsp/

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Main Contents

• Important types of signals and their operations
• Linear and shift-invariant system
• Easier to analyze and implement
• The convolution and difference equation
  representations
• Representations and implementation of signal and
  systems using MATLAB

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Discrete-time signals

• Analog and discrete signals
• analog signal
• t represents any physical quantity, time in sec.
• Discrete signal discrete-time signal
• N is integer valued, represents discrete
  instances in times

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Discrete-time signal

• In Matlab, a finite-duration sequence
  representation requires two vectors, and each for
  x and n.
• Example
• Question whether or not an arbitrary
  infinite-duration sequence can be represented in
  MATLAB?

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Types of sequences

• Elementary sequence for analysis purposes
• 1. Unit sample sequence
• Representation in MATLAB

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Function x,nimpseq(n0,n1,n2)

• A nn1n2
• x zeros(1,n2-n11) x(n0-n11)1
• B nn1n2 x (n-n0)0 stem(n,x,ro)

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2. Unit step sequence
A nn1n2 xzeros(1,n2-n21)
x(n0-n11end)1 B nn1n2 x(n-n0)gt0
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3. Real-valued exponential sequence
For Example
n010 x(0.9).n stem(n,x,ro)
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4. Complex-valued exponential sequence
Attenuation ???? frequency in radians For
Example n010 xexp((23j)n)
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5. Sinusoidal sequence
Phase in radians For Example n010
x3cos(0.1pinpi/3)2sin(0.5pin)
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6. Random sequence

• Rand(1,N)
• Generate a length N random sequence whose
  elements are uniformly distributed between 0,1
• Randn(1,N)
• Generate a length N Gaussian random sequence with
  mean 0 and variance 1. en 0,1

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7. Periodic sequence

• A sequence x(n) is periodic if x(n)x(nN)
• The smallest integer N is called the fundamental
  period
• For example
• A xtildex,x,x,x
• B xtildexones(1,P) xtildextilde()
  xtildextilde transposition

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Operations on sequence

• 1. Signal addition
• Sample-by-sample addition
• x1(n)x2(n)x1(n)x2(n)

Function y,nsigadd(x1,n1,x2,n2) nmin(min(n1),m
in(n2)) max(max(n1),max(n2)) y1zeros(1,length(n
)) y2y1 y1(find((ngtmin(n1))
(nltmax(n1))1))x1 y2(find((ngtmin(n2))
(nltmax(n2))1))x2 Yy1 y2
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2. Signal multiplication

• Sample-by-sample multiplication
• Dot multiplication
• x1(n).x2(n)x1(n) x2(n)

Function y,nsigmult(x1,n1,x2,n2) nmin(min(n1),
min(n2)) max(max(n1),max(n2)) y1zeros(1,length
(n)) y2y1 y1(find((ngtmin(n1))
(nltmax(n1))1))x1 y2(find((ngtmin(n2))
(nltmax(n2))1))x2 Yy1 . y2
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3. Scaling

• ax(n)ax(n)

4. Shifting

• y(n)x(n-k)
• mn-k yx

5. folding

• y(n)x(-n)
• yfliplr(x) n-fliplr(n)

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6. Sample summation ss
sum(x(n1n2) 7. Sample production sp
prod(x(n1n2))
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8. Signal energy se sum(x . conj(x))
or se sum(abs(x) . 2) 9. Signal power
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Examples

• Ex020100 composite basic sequences
• Ex020200 operation on sequences
• Ex020300 complex sequence generation
• Ex020400 even-odd decomposition

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Some useful results

• Unit sample synthesis
• Any arbitrary sequence can be synthesized as a
  weighted sum of delayed and scaled unit sample
  sequence.
• Even and odd synthesis
• Even (symmetric) xe(-n)xe(n)
• Odd (antisymmetric) xo(-n)-xo(n)
• Any arbitrary real-valued sequence can be
  decomposed into its even and odd component x
  (n)xe(n) xo(n)

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Function xe, x0, m evenodd(x,n) If
any(imag(x) 0) error(x is not a real
sequence) End m -fliplr(n) m1 min(m,n)
m2 max(m,n) mm1m2 nm n(1)-m(1) n1
1length(n) x1 zeros(1, length(m)) x1(n1nm)
x x x1 xe 0.5 (x flipflr(x)) xo
0.5(x - fliplr(x))
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The geometric series

• A one-side exponential sequence of the form an,
  ngt0, where a is an arbitrary constant, is
  called a geometric series.
• Expression for the sum of any finite number of
  terms of the series

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Correlations of sequences

• It is a measure of the degree to which two
  sequences are similar. Given two real-valued
  sequences x(n) and y(n) of finite energy,
• Crosscorrelation
• Autocorrelation

The index l is called the shift or lag parameter.
The special case y(n)x(n)
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Discrete Systems

• Mathematically, an operation T.
• y(n) T x(n)
• x(n) excitation, input signal
• y(n) response, output signal
• Classification
• Linear systems
• Nonlinear systems

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Linear operation L.

• Iff L. satisfies the principle of superposition
• The output y(n) of a linear system to an
  arbitrary input x(n)
• is called impulse response, and is
  denoted by h(n,k)

h(n,k) the time-varying impulse response
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Linear time-invariant (LTI) system

• A linear system in which an input-output pair is
  invariant to a shift n in time is called a linear
  times-invariant system
• y(n) Lx(n) ---? y(n-k) Lx(n-k)
• The output of a LTI system is call a linear
  convolution sum
• An LTI system is completely characterized in the
  time domain by the impulse response h(n).

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Properties of the LTI system

• Stability
• A system is said to be bounded-input
  bounded-output (BIBO) stable if every bounded
  input produces a bounded output.
• Condition absolutely summable
• To avoid building harmful systems or to avoid
  burnout or saturation in system operation

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Properties of the LTI system

• Causality
• A system is said to be causal if the output at
  index n0 depends only on the input up to and
  including the index n0
• The output does not depend on the future values
  of the input
• Condition h(n) 0, n lt 0
• Such a sequence is termed a causal sequence.
• To make sure that systems can be built.

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Convolution

• Convolution can be evaluated in many different
  ways
• If the sequences are mathematical functions, then
  we can analytically evaluate x(n)h(n) for all n
  to obtain a functional form of y(n)
• Graphical interpretation, folded-and-shifted
  version
• Matlab implementation
• Function y,nyconv_m(x,nx,h,nh)
• nyb nx(1)nh(1) nye nx(length(x))nh(length(h
  ))
• ny nybnye
• n conv(x,h)

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Function form of convolution
Three different conditions under which u(n-k) can
be evaluated Case 1 nlt0 the nonzero
values of x(n)and y(n) do not overlap. Case 2
0ltnlt9 partially overlaps Case 3 ngt9
completely overlaps
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Folded-and-shifted

• Signals xx(1),x(2),x(3),x(4),x(5)
• System Impulse Response hh(1),h(2)h(3),h(4)
• yconv(x,h)
• y(1)x(1)h(1) y(2)x(1)h(2)x(2)h(1)
• y(3)x(1)h(3)x(2)h(2)x(3)h(1)
• x(1),x(2),x(3),x(4),x(5)
  
• h(4),h(3),h(2),h(1)

Note that the resulting sequence y(n) has a
longer length than both the x(n) and h(n)
sequence.
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Sequence correlations revisited

• The correlation can be computed using the conv
  function if sequences are of finite duration.
• Example 2.8
• The meaning of the crosscorrelation
• This approach can be used in applications like
  radar signal processing in identifying and
  localizing targets.

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Difference Equation

• An LTI discrete system can also be described by a
  linear constant coefficient difference equation
  of the form
• If aN 0, then the difference equation is of
  order N
• It describes a recursive approach for computing
  the current output,given the input values and
  previously computed output values.

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Solution of difference equation

• y(n) yH(n) yP(n)
• Homogeneous part yH(n)
• Particular part yP(n)
• Analytical approach using Z-transform will be
  discussed in Chapter 4
• Numerical solution with Matlab
• y filter(b,a,x)
• Example 2.9

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Zero-input and Zero-state response

• In DSP the difference equation is generally
  solved forward in time from n0. Therefore
  initial conditions on x(n) and y(n) are necessary
  to determine the output for ngt0.
• Subject to the initial conditions

Solution
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Zero-input and Zero-state response

• yZI(n) zero-input solution
• A solution due to the initial conditions alone
• yZS(n) zero-state solution
• A solution due to input x(n) alone

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Digital filter

• Discrete-time LTI systems are also called digital
  filter.
• Classification
• FIR filter IIR filter
• FIR filter
• Finite-duration impulse response filter
• Causal FIR filter
• h(0)b0,,h(M)bM
• Nonrecursive or moving average (MA) filter
• Difference equation coefficients, bm and a01
• Implementation in Matlab Conv(x,h) filter(b,1,x)

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IIR filter

• Infinite-duration impulse response filter
• Difference equation
• Recursive filter, in which the output y(n) is
  recursively computed from its previously computed
  values
• Autoregressive (AR) filter

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ARMA filter

• Generalized IIR filter
• It has two parts MA part and AR part
• Autoregressive moving average filter, ARMA
• Solution
• filter(b,a,x) bm, ak

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Reference and Assignment

• Textbook pp1 to pp35
• Chinese reference book pp1 to pp18
• ???????????????(???),?????,2001?1?
• Exercises
• Textbook p2.1b,c p2.2b,d ?2.5
• Textbook P2.12b, 2.15, 2.17b, ?2.8