Chapter2. DISCRETE-TIME SIGNALS AND SYSTEMS
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Chapter2. DISCRETE-TIME SIGNALS AND SYSTEMS. 2.0 Introduction 2.1 Discrete-Time Signals : Sequences 2.2 Discrete-Time Systems 2.3 Linear Time-Invariant Systems 2.4 Properties of Linear Time-Invariant Systems 2.5 Linear Constant-Coefficient Difference Equations

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Chapter2. DISCRETE-TIME SIGNALS AND SYSTEMS

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Chapter2 discrete time signals and systems

Chapter2. DISCRETE-TIME SIGNALS AND SYSTEMS

2.0 Introduction

2.1 Discrete-Time Signals : Sequences

2.2 Discrete-Time Systems

2.3 Linear Time-Invariant Systems

2.4 Properties of Linear Time-Invariant Systems

2.5 Linear Constant-Coefficient Difference Equations

2.6 Frequency-Domain Representation

2.7 Representation of Sequences of the Fourier Transform

2.8 Symmetry Properties of the Fourier Transform

2.9 Fourier Transform Theorems

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Chapter2 discrete time signals and systems

2.1.Discrete - Time Signals

x[n]= x(t)|t=nT

n : -1,0,1,2,…

T: sampling period

x(t) : analog signal

ii) unit step sequence

u[n] = 1,n0

0,n0

i) unit impulse signal(sequence)

d[n] = 1,n=0

0,n0

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Chapter2 discrete time signals and systems

iii) exponential/sinusoidal sequence

x[n]= Aej(won+), Acos(won+)

- not necessarily periodic in n with period 2p/wo

- periodic in nwithperiod N (discrete number)

for woN=2pk or wo= 2pk/N

[note] x(t)= Ae j(wo t+)

is periodic in t with period T= 2p/wo (continuous value)

iv) general expression

x[n] = Sx[k]d[n-k]

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Chapter2 discrete time signals and systems

x[n]

y[n]

T[ ]

2.2Discrete-Time Systems

System : signal processor

i) memoryless or with memory

y[n] = f(x[n]), y[n]=f(x[n-k]) with delay

ii) linearity

x1[n]y1[n]

x2[n]y2[n]

a1x1[n] + a2x2[n] a1y1[n] + a2 y2[n]

- e.g. T[a1x1[n] + a2x2[n]] = T[a1x1[n]] + T[a2x2[n]]

= a1T[x1[n]] + a2T[x2[n]]

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Chapter2 discrete time signals and systems

iii) time-invariance

x[n]  y[n] x[n-n0]  y[n-n0]

- e.g., T[x[n-n0]] = T[x[n]] | n  n-no

- e.g., d[n]  h[n]  d[n-k]  h[n-k]

- counter-example : decimator T[ ] = x[Mn]

iv) causality

y[n] for n=n1, depends on x[n] for nn1 only

- counter-example : y[n] = x[n+1] - x[n]

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Chapter2 discrete time signals and systems

v) stability

bounded input yields bounded output(BIBO)

|x[n]| <  for all n  |y[n]| <  for all n

- counter-example : y[n] = S u[k] = 0,n<0

n+1,n0

unbounded ( no fixed value By exists that keeps

y[n]  By <  .)

n

k=-

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Chapter2 discrete time signals and systems

In general, let x[n] = Sx[k]d[n-k]

In general, let x[n] = Sx[k]d[n-k]

y[n] = T[ Sx[k]d[n-k]]

k=-

k

2.3 Linear Time-Invariant Systems

d[n]

h[n]

LTI

T[d[n]] : impulse response

y[n]

x[n]

( by linearity)

= Sx[k]T[d[n-k]]

k

( by time-invariance)

= Sx[k]h[n-k]

coefficient

= x[n]*h[n]

= Sx[n-r]h[r]

r=-

Convolution!

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Chapter2 discrete time signals and systems

d[n] = 1,n=0

0,n0

Sh[k]x[n-k]

k=0

In summary,

h[n]

x[n]

y[n]

y[n] = x[n]*h[n]

LTI

h[n] : unique characteristic of the LTI system

- causal LTI system

y[n] =

Sh[k]x[n-k] =

k=-

[note] h[n] = T[d[n]] = 0 n<0. as

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Chapter2 discrete time signals and systems

- Stable LTI System

S | x[n-k] | • | h[k] |

Sh[k]x[n-k] | 

|y[n]| = |

k=-

k=-

S| h[k] |

< 

 By

 Bx

k=-

S| h[k] |

Therefore,

< 

k=-

In fact, this is necessary and sufficient condition for stability of a BIBO system.

( You prove it! )(*1)

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Chapter2 discrete time signals and systems

~

~

~

x[n]

y[n]

y[n]

- Example of non-LTI system - Decimator

x[n]

Decimator

M

y[n] = x[Mn]

?

= x[n-1]

= y[n-1] = x[M[n-1]]

M=3

y[n] = x[Mn]

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Chapter2 discrete time signals and systems

~

y[n]

y[n]

y[n-1]

No!

= x[Mn-1]

 x[M[n-1]] = y[n-1]

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Chapter2 discrete time signals and systems

2.4 Properties of LTI System

h[n]

x[n]

y[n] = x[n]*h[n]

LTI

i) parallel connection

h[n] = h1[n] + h2[n]

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Chapter2 discrete time signals and systems

ii) cascade connection

h[n] = h1[n]*h2[n]

=h2[n]* h1[n]

[note] distinctive feature of digital LTI system (*2)

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Chapter2 discrete time signals and systems

M

M

= Sbrx[n-r]

= Sbrx[n-r]

r = 0

r = 0

N

Saky[n-k]

k = 0

2.5 Linear Difference Equations

LTI

x[n]

y[n]

i) Case 1 : N=0  FIR System

(set a0 =1, for convenience)

y[n]

For impulse input, x[n]=d[n], the response is

h[n]=0,n<0 or n>M

br0 n  M finite impulse response!

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Chapter2 discrete time signals and systems

M

= Sbrx[n-r]

r = 0

N

Saky[n-k]

k = 1

ii) Case 2 : N  0  IIR System

(set a0 =1, for convenience)

y[n]

e.g., set N=1 (lst order), and a1 = -a

 y[n] = b0 x[n] + ay[n-1]

-

For impulse input x[n] = d[n], the response is

1) If assume a causal system, i.e., y[n]=0 n<0.

y[0] = b0 d[0] + ay[-1] = b0

y[1] = b0 d[1] + ay[0] = ab0

• • •

y[n] = b0 d[n] + ay[n-1] = anb0

h[n] = anb0u[n]

infinite impulse response!

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Chapter2 discrete time signals and systems

2) If assume an anti-causal system, i.e., y[n]=0 n>0.

y[n-1] = a-1(-b0 d[n] + y[n])

y[0] = a-1(-b0 d[1] + y[1]) = 0

y[-1] = a-1(-b0 d[0] + y[0]) = a-1b0 h[n] = -anb0u[-n-1]

• • •

y[-n] = a-1(- b0 d[n+1] + y[n+1]) = -anb0

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Chapter2 discrete time signals and systems

^

^

y = Ax=lx

y[n] = Sh[k] ejw(n-k)

^

^

x

x

k = -

2.6 Frequency-Domain Representation

• Linear System

x

y = Ax

A

scalar, eigenvalue

for eigenvector input

• LTI System

x[n]

y[n]=x[n]*h[n]

h[n]

y[n]=ejwn*h[n]

ejwn

= H(ejw)ejwn

Fourier transform

( Sh[k] e-jwk )ejwn =H(ejw) ejwn

=

k = -

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Chapter2 discrete time signals and systems

Sh[k] e-jwk

k = -

p

h[k] = 1/2p H(ejw) ejwkdw

-p

• Fourier Transform

H(ejw) =

You prove this! (*3)

• Condition for existence of FT

| X(ejw) | <   S | x[n] | < 

“ absolutely summable”

(BIBO stable condition)

• Real - imaginary

H(ejw) = HR(ejw) + j HI(ejw)

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Chapter2 discrete time signals and systems

• Magnitude-phase

H(ejw) = | H(ejw) |ej H(ejw)

(e.g.) ideal delay system

x[n]

y[n] = x[n-nd]

h[n]

y[n] = ejw(n-nd)= H(ejw) ejwn

H(ejw) = e-jwnd

ejwn

HR(ejw) = coswnd HI(ejw) = - sinwnd

| H(ejw) | = 1 H(ejw) = -wnd

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Chapter2 discrete time signals and systems

(e.g.) sinusoidal input

x[n] = Acos(w0n + j) = (A/2) ejjejw0n + (A/2) e-jje-jw0n

y[n] = H(ejw0) (A/2) ejjejw0n + H(e-jw0) (A/2) e-jje-jw0n

= (A/2)(H(ejw0) ejjejw0n + H(e-jw0) e-jje-jw0n)

y[n] = H(ejw) ejwn

<1>

= (A/2){ (H(ejw0) ejjejw0n ) + (H(ejw0) ejjejw0n)* }

= ARe{H(ejw0) ejjejw0n}

= A | H(ejw0) |(cosw0n + j + q)

= A cos (w0(n-nd) + j)

<2>

<3>

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Chapter2 discrete time signals and systems

= ( Sh*[n] e-jw0n)* = H*(ejw0)

n

<1> if h[n] real

h[n] = hR[n]+jhI[n] = hR[n] = h*[n]

H(e-jw0) =

Sh[n] e-(-jw0n)

n

<2> H(ejw) = | H(ejw) |ejq

<3> if ideal delay system

with | H(ejw) | = 1,  H(ejw) = q = -wnd

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Chapter2 discrete time signals and systems

(e.g.) ideal lowpass filter (LPF)

x[n]

y[n]

1

-wc

wc

Hl(ejw) = 1. e-jwnd|w| 

0.elsewhere

wc

periodic with period 2p

Inputx[n] = Acos(w0n + j)

output y[n] = Acos(w0(n-nd )+ j) , if

0 ,otherwise

wo

< wc

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Chapter2 discrete time signals and systems

sinw0[n-nd ]

p[n-nd ]

wc

hl [k] = e-jwnd ejwndw

-wc

(e.g.) Fourier transform of anu[n]|a|<1

X(ejw)

= Sane-jwn = S (ae-jw)n

=

n = 0

n = 0

(e.g.) inverse Fourier transform of ideal LPF

=

- <n < 

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Chapter2 discrete time signals and systems

- Data length vs. Spectrum Change

- Gibb’s phenomenon (page 52)

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Chapter2 discrete time signals and systems

- Error reduces in RMS sense

but not in Chebyshev sense.

 Limitation of rectangular windowing

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Chapter2 discrete time signals and systems

2.8 Symmetry Properties (table 2.1)

i) even / odd

e : conjugate symmetric  even

o : conjugate anti-symmetric  odd

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Chapter2 discrete time signals and systems

ii) real/imaginary

iii) conjugation/reversal

S

= ( S )*

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Chapter2 discrete time signals and systems

iv) real/imaginary - even/odd

v) for real x[n]

real  even

imaginary  odd

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Chapter2 discrete time signals and systems

(e.g.) x[n] = anu[n]|a|<1, real (example 2.25)

even

odd

even

odd

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Chapter2 discrete time signals and systems

Dashed line : a = 0.5Solid line : a = 0.9

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Chapter2 discrete time signals and systems

2.9 Fourier Transform Theorems (table 2.2)

i) linearity

ii) time shifting

iii) frequency shifting

iv) time reversal

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Chapter2 discrete time signals and systems

v) differentiation in frequency

vi) Parserval’s relation

vii) convolution relation

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Chapter2 discrete time signals and systems

viii) modulation/windowing relation

ix) fundamental functions

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Chapter2 discrete time signals and systems

1.

0.

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Chapter2 discrete time signals and systems

H.W. of Chapter 2

Matlab: [1] Consider the following discrete-time systems characterized by the difference equations:

y[n]=0.5x[n]+0.27x[n-1]+0.77x[n-2]

Write a MATLAB program to compute the output of the above systems for an input x[n]=cos(20n/256)+cos(200n/256), with 0n<299 and plot the output.

[2] The Matlab command y=impz(num,den,N) can be used to compute the first N samples of the impulse response of the causal LTI discrete-time system.

Compute the impulse response of the system described by

y[n]-0.4y[n-1]+0.75y[n-2]

=2.2403x[n]+2.4908x[n-1]+2.2403x[n-2]

and plot the output using the stem function.

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Chapter2 discrete time signals and systems

H.W. of Chapter 2

Text: [3]2-15 [4]2-30 [5]2-42 [6]2-56 [7]2-58

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