1497-Chapter2

Chapter 2Discrete-Time Signals and Systems

Der-Feng Tseng

Department of Electrical EngineeringNational Taiwan University of Science and Technology

(through the courtesy of Prof. Peng-Hua Wang of National Taipei University)

February 19, 2015

Der-Feng Tseng (NTUST) DSP Chapter 2 February 19, 2015 1 / 61

Outline

1 2.1 Discrete-Time Signals: Sequences

2 2.2 Discrete-Time Systems

3 2.3 Linear Time-Invariant Systems

4 2.4 Properties of Linear Time-Invariant Systems

5 2.5 Linear Constant-Coefficient Difference Equations

6 2.6 Frequency-Domain Representation

7 2.7 Representation of Sequences by Fourier Transform

8 2.8 Symmetry Properties of the Fourier Transform

9 2.9 Fourier Transform Theorems

10 2.10 Discrete-Time Random Random Signals


2.1 Discrete-Time Signals: Sequences


Definition

A sequence of number x, in which the nth number in the sequence isdenoted x[n] is written as x = {x[n]}, < n

Basic sequences 1/3

The unit sample sequence is defined by

[n] =

{0, n 6= 0

1, n = 0.

Any sequence can be expressed as a sum of scaled, delayed impulse.

The unit step sequence is defined by

u[n] =

{1, n 0

0, n < 0.

The exponential sequence is x[n] = An.


Basic sequences 2/3

Any sequence can be expressed as a sum of scaled, delayed impulse

p[n] =

k=

p[k][n k]

u[n] can be expressed as a sum of delayed impulse

u[n] =

k=0

[n k], or u[n] =

nk=

[k]

Example 2.1 An exponential sequence x[n] that is zero for n < 0 can bewritten as x[n] = Anu[n].


Basic sequences 3/3

The complex exponential sequence is x[n] = Aej(0n+).

The sinusoidal sequence is x[n] = A cos(0n+ ) where 0 is call thefrequency and is called the phase.

Difference between continuous and discrete complex exponentials

fi(t) = ejit : f1(t) = f2(t) 1 = 2

xi[n] = ejin : x1[n] = x2[n] 1 2 = 2r

where i = 1, 2 and r is an integer.


Basic Operations

Delay/shift: x[n n0] is delayed sequence of x[n] where n0 is aninteger.

Periodic sequence: x[n] = x[n+N ] where N is the period.

Difference between continuous and discrete complex exponentials

f(t) = ej0t is periodic with period 2/0

x[n] = ej0n is periodic with period N if 0N = 2k

where k is an integer.

Example 2.2 What is the period of x1[n] = cos(n/4), x2[n] =cos(3n/8), and x3[n] = cos(n)?


2.2 Discrete-Time Systems


Definition

A system is defined as a transformation or operator that maps aninput x[n] to an output y[n].

y[n] = T{x[n]}

Example 2.3 The ideal delay system is defined by y[n] = x[n nd] wherend is a positive integer. If nd < 0, this is a time-advancesystem.

Example 2.4 The moving average system is defined by

y[n] = (x[n+M1] + x[n+M1 1] + + x[n]

+ x[n 1] + + x[nM2]) /(M1 +M2 + 1)


Properties 1/4

Memoryless: y[n] depends on only x[n], not x[n n0] for n0 6= 0.

Example 2.3 y[n] = x[n nd] is not memoryless unless nd = 0.

Example 2.4 The moving-average system is not memoryless unlessM1 = M2 = 0.

Example 2.5 y[n] = (x[n])2 is memoryless.


Properties 2/4

Additivity: T{x1[n] + x2[n]} = T{x1[n]}+ T{x2[n]}.

Homogeneity/scaling: T{ax1[n]} = aT{x1[n]}.

A system with additivity but not homogeneity: T {x[n]} = {x[n]}

Linearity/superposition:T{ax1[n] + bx2[n]} = aT{x1[n]}+ bT{x2[n]}.

Example 2.6 The accumulator is linear

y[n] = x[n] + x[n 1] + =n

k=

x[k] =k=0

x[n k]

Example 2.7 w[n] = log10(|x[n]|) is nonlinear.


Properties 3/4

Time-invariant: If y[n] = T{x[n]} then y[n n0] = T{x[n n0]}.

Example 2.8 The accumulator is time-invariant.

Example 2.9 The compressor y[n] = x[Mn] is not time-invariant where Mis a positive integer.

Causality: y[n] = T{x[n]} depends on only x[n], x[n 1], . . .

Example 2.10 Forward difference y[n] = x[n+ 1] x[n] is not causal.Backward difference y[n] = x[n] x[n 1] is causal.


Properties 4/4

BIBO stability: Every bounded input sequence produce a boundedoutput sequence.

Example 2.11 The accumulator is not stable.


2.3 Linear Time-Invariant Systems


LTI Systems

Linear time-invariant (LTI) system.

If a system y[n] = T{x[n]} is LTI, then it can be characterized byh[n] = T{[n]} in the sense

y[n] =

k=

x[k]h[n k] , x[n] h[n]

The above equation is called the convolution sum. h[n] is called theimpulse response of the system.


Proof of Convolution

y[n] = T{x[n]}

= T

{

k=

x[k][n k]

}(representation)

=

k=

x[k]T{[n k]} (linearity)

=

k=

x[k]h[n k] (time-invariant)


Example 2.13

h[n] = u[n] u[nN ] = 1, 0 n N 1.

x[n] = anu[n]

h[n] x[n] =

k=

x[k]h[n k] =

k=

h[k]x[n k]

=

N1k=0

anku[n k]

=

0, n < 0,1an+1

1a , 0 n N 1,

anN+1(1aN

1a

), n > N 1.


Convolution by Tabular

Compute {x[2], x[1], x[0], x[1]} {h[1], h[0], h[1]}.

x2 x1 x0 x1 h1 h0 h1

h1x2 h1x1 h1x0 h1x1h0x2 h0x1 h0x0 h0x1

+ h1x2 h1x1 h1x0 h1x1y3 y2 y1 y0 y1 y2

Prob. 2.22a Compute {0, 1} {2, 1}.

Prob. 2.22b Compute {2,1} {1, 2, 1}.


2.4 Properties of Linear Time-Invariant Systems


Properties of LTI Systems

Commutative: x[n] h[n] = h[n] x[n]

Distributive: x[n] (h1[n] + h2[n]) = x[n] h1[n] + x[n] h2[n]

Cascade: h1[n] h2[n]

Parallel connection: h1[n] + h2[n]

Finite-duration impulse response (FIR) systems have only finitenonzero impulse response.

Infinite-duration impulse response (IIR) systems have infinite nonzeroimpulse response.

Causality: h[n] = 0 for n < 0


BIBO stability

An LTI system is stable if and only if

k=

|h[k]| = S

Examples 1/2

Example 2.3 Ideal delay system: h[n] = [n nd] (FIR, stable)

Example 2.4 Moving average

h[n] =1

M1 +M2 + 1([n +M1] + [n +M1 2] +

+ [n] + [n 1] + + [n M2])

Example 2.5 y[n] = (x[n])2, not LTI systems.

Example 2.6 Accumulator

h[n] = [n] + [n 1] + = u[n]

(IIR, not stable)


Examples 1/2

Example 2.7 w[n] = log10(|x[n]|) not LTI systems.

Example 2.9 Compressor: y[n] = x[Mn] not LTI systems.

Example 2.10 Forward difference: y[n] = [n + 1] [n].

Example 2.10 Backward difference: y[n] = [n] [n 1]. Backwarddifference is the inverse system of accumulator.

u[n] ([n] [n 1]) = u[n] u[n 1] = [n]


2.5 Linear Constant-Coefficient DifferenceEquations


LCC Difference Equations

An N th-order linear constant-coefficient (LCC) difference equations

Nk=0

aky[n k] =

Mm=0

bmx[nm]

Example 2.14 Difference equation representation of the accumulator.

y[n] =

nk=

x[k]

Example 2.15 Difference equation representation of a moving-averagesystem.

y[n] =1

M2 + 1

M2k=0

x[n k]


Solve LCC DE 1/2

For the difference equation

Nk=0

aky[n k] =

Mm=0

bmx[nm],

the solution can be written by y[n] = yh[n] + yp[n].yh[n] is the solution to x[n] = 0, called the homogeneous solution. Theassociated homogeneous equation is

Nk=0

aky[n k] = 0.


Solve LCC DE 2/2

Let y[n] = zn, we haveNk=0

akzk = 0

Denote its solution as z1, z2, . . . , zk. Then

yh[n] =

Nk=1

akznk .


Example 2.16

Solve y[n] = ay[n 1] + x[n], x[n] = K[n], y[1] = c

Solution. For n 0,

y[0] = ay[1] + x[0] = ac+K

y[1] = ay[0] + x[1] = a(ac+K) = a2c+ aK

For n < 0, y[n 1] = a1(y[n] x[n]), we have

y[2] = a1(y[1] x[1]) = a1c

y[3] = a1(y[2] x[2]) = a2c

Therefore, y[n] = an+1c+Kanu[n].

This system is NOT causal.

This system is NOT linear.

This system is NOT time-invariant.


Initial-Rest Conditions

We can convert an LTI system to an LCC difference equation.

We may not obtain an LTI system from an LCC difference equation.

We may obtain several LTI systems from a LCC difference equation.

Linearity, time invariance, and causality of system will depend on theauxiliary conditions.

Initial-rest conditions. If x[n] = 0 for n < n0, then y[n] = 0 forn < n0. The system is LTI and causal for initial rest conditions.


2.6 Frequency-Domain Representation


Eigenfunctions

Suppose that an LTI system with impulse response h[n] have its inputx[n] = ejn, < n

Example 2.17

The frequency response of the ideal delay system y[n] = x[n nd] isH(ej) = ejd . We have HR(e

j) = cos(nd), HI(ej) = sin(nd).

The magnitude response |H(ej)| = 1 and phase responseH(ej) = nd.


Linearity

If x[n] =k

kejkn, then y[n] =

k

kH(ejk)ejkn.

Example 2.18

x[n] = A cos(0n+ ) =A

2ejej0n +

A

2ejej0n

y[n] =A

2ejH(ej0)ej0n +

A

2ejH(ej0)ej0n

If h[n] is real, it can be shown that H(ej0) = H |ast(ej0), and

y[n] = A|H(ej0)| cos(0n+ + H(ej0)


Example 2.20

The frequency response of the moving-average system is

H(ej) =1

M1 +M2 + 1

M2n=M1

ejn =ej(M2M1)/2

M1 +M2 + 1

sin[(M1 +M2 + 1)/2]

sin(/2)


Causal Input 1/2

An input x[n] = ejnu[n] is applied to an LTI system and generates theoutput y[n] = yss[n] + yt[n] where yss[n] = H(e

j)ejn is thesteady-state response and yt[n] is the transient response.

y[n] =

k=

h[k]x[n k]

=

nk=

h[k]ej(nk)u[n k]

=

k=

h[k]ej(nk)

yss[n]

k=n+1

h[k]ej(nk)

yt[n]


Causal Input 2/2

yt[n] =

k=n+1

h[k]ejkejn

If h[n] = 0 except 0 n M , then yt[n] = 0 for n > M 1.

If h[n] has infinite duration, then

|yt[n]|

k=n+1

|h[k]|

k=0

|h[k]|

If the system is stable, |yt[n]| is bounded.

In fact, if the system is stable, then

|yt[n]|

k=n+1

|h[k]| 0

by Cauchy condition.


2.7 Representation of Sequences by FourierTransform


Fourier Transform of Sequences

X(ej) =

n=

x[n]ejn

x[n] =1

2

X(ej)ejnd

X(ej) = XR(ej) + jXI(e

j)

X(ej) = |X(ej)|ejX(ej)

X(ej) has a period 2.ARG[X(ej)] = X(ej), X(ej) < . ARG[X(ej)] maybe not continuous.If we want a continuous phase functions for 0 < < , we use thenotation arg[X(ej)].


Convergence 1/2

If x[n] is absolute summable, i.e.,

n=

|x[n]|

Convergence 2/2

If x[n] is square summable, i.e.,

n=

|x[n]|2

Example 2.22

Hlp(ej) =

{1, || < c,

0, otherwise, hlp[n] =

sincn

n, < n

Example 2.22


Example 2.23 & 2.24

Example 2.23 Impulse train

x[n] = 1, X(ej) =

r=

2( + 2r)

= 2(), < .

Note that X(ej) is periodic with period of 2.

Example 2.24 Complex exponential sequences

x[n] = ej0n, X(ej) =

r=

2( 0 + 2r)


Example: Fourier Transform of Unit Step Function

x[n] = u[n], X(ej) =1

1 ej+

r=

( + 2r)

Proof. Let x[n] = 12 + s[n] where s[n] =12 for n 0 and s[n] =

12 for

n < 0. Then the Fourier transform of x[n] is the sum of the Fouriertransform of a constant 12 and that of s[n]. Consider

t[n] =

{an, n 0

an, n < 0., |a| < 1.

We have s[n] = 12 lima1t[n] and S(ej) =

1

2lima1

T (ej).


Example: Fourier Transform of Unit Step Function

It is easy to evaluate T (ej)

T (ej) =

n=0

anejn

1n=

anejn =1 + a2 2aej

(1 aej)(1 aej)

and

lima1

T (ej) =2

1 ej

Therefore,

X(ej) =1

1 ej+ (), < .

Remark. The () in X(ej) represents the effect of DC component inx[n].


2.8 Symmetry Properties of the Fourier Transform


Even-Odd Decomposition 1/2

For a complex sequence x[n]

conjugate-symmetric: x[n] = x[n].conjugate-antisymmetric: x[n] = x[n].

For a real sequence x[n]

even: x[n] = x[n].odd: x[n] = x[n].

Any complex (real) sequence can be represented as a sum of aconjugate-symmetric (even) sequence and a conjugate-antisymmetric(odd) sequence: x[n] = xe[n] + xo[n] where

xe[n] =1

2(x[n] + x[n])

xo[n] =1

2(x[n] x[n]).


Even-Odd Decomposition 2/2

Any complex (real) Fourier transform X(ej) can be represented as a sumof a conjugate-symmetric (even) function Xe(e

j) and aconjugate-antisymmetric (odd) function Xo(e

j)

X(ej) = Xe(ej) +Xo(e

j)

where

Xe(ej) =

1

2

[X(ej) +X(ej)

]Xo(e

j) =1

2

[X(ej)X(ej)

].


Properties

If x[n]F X(ej), then

1 x[n]F X(ej)

2 x[n]F X(ej)

3 {x[n]}F Xe(e

j)

4 j{x[n]}F Xo(e

j)

5 xe[n]F {X(ej)}

6 xo[n]F j{X(ej)}


Properties

If x[n] is real and x[n]F X(ej), then

1 Fourier Transform is conjugate symmetric. X(ej) = X(ej)

2 Real part is even. {X(ej)} = {X(ej)}

3 Imaginary part is odd. {X(ej)} = {X(ej)}

4 Magnitude is even. |X(ej)| = |X(ej)|

5 Phase is odd. X(ej) = X(ej)

6 xe[n]F {X(ej)}

7 xo[n]F j{X(ej)}


2.9 Fourier Transform Theorems


Theorems 1/2

Linearity

ax1[n] + bx2[n]F aX1(e

j) + bX2(ej)

Time shiftx[n nd]

F ejndX(ej)

Frequency shift

ej0nx[n]F X(ej(0))

Time-reversalx[n]

F X(ej)


Theorems 2/2

Differentiation in Frequency

nx[n]F j

dX(ej)

d

Parsevals Theorem

n=

|x[n]|2 =1

2

|X(ej)|2d

Convolution Theoremx[n] h[n]

F X(ej)H(ej)

Modulation Theorem

x[n]w[n]F

1

2

X(ej)W (ej())d


Examples 2.26 2.29

Example 2.26 Find F{anu[n 5]}.

Example 2.27 Find inverse Fourier transform of

X(ej) =1

(1 aej)(1 aej)

Example 2.28 Find inverse Fourier transform of

H(ej) =

{ejnd , c < || < ,

0, || < c.

Example 2.29 Find impulse response of the following stable, LTI system

y[n]1

2y[n 1] = x[n]

1

4x[n 1].


2.10 Discrete-Time Random Random Signals


I-O Relationships 1/2

Let the input sequence x[n] be a random precess with mean functionmx[n] , E{x[n]} and autocorrelation function

xx[n, n+ k] , E{x[n]x[n + k]}.

A wide-sense stationary (WSS) random process has meanindependent of n

mx[n] = mx

and has autocorrelation function depending on time difference only

xx[n, n+ k] = xx[k].

Suppose that a stable LTI system is characterized by

y[n] =

k=

h[n k]x[k] =

k=

h[k]x[n k].

When the input sequence x[n] is a WSS random precess, is theoutput y[n] also a WSS random process?


I-O Relationships 2/2

The mean of y[n] is

my[n] = mx

k=

h[k] = mxH(ej0) = my

The autocorrelation function yy[m] is

yy[n, n+m] = E{y[n]y[n +m]} =

=

xx[m ]chh[]

where

chh[] =

k=

h[k]h[ + k]


Power Density Spectrum 1/2

Assume that mx = 0. Let xx(ej) = F{xx[m]}, yy(e

j) =F{yy [m]}, and Chh(e

j) = F{chh[]}. We have

yy(ej) = Chh(e

j)xx(ej)

Chh(ej) = H(ej)H(ej) = |H(ej)|2

yy(ej) = xx(e

j)|H(ej)|2

xx(ej) is called the power density spectrum of x[n] because

total power = E{x2[n]} = xx[0]

=1

2

xx(ej)d


Power Density Spectrum 2/2

Since x[n] is real, we have

xx(ej) is real and even because xx[m] is even and real.

xx(ej) 0.


Example 2.30

A white noise is a signal for which xx[m] = 2x[m].

xx(ej) = 2x.

yy(ej) = 2x|H(e

j)|2


2.1 Discrete-Time Signals: Sequences2.2 Discrete-Time Systems2.3 Linear Time-Invariant Systems2.4 Properties of Linear Time-Invariant Systems2.5 Linear Constant-Coefficient Difference Equations2.6 Frequency-Domain Representation2.7 Representation of Sequences by Fourier Transform2.8 Symmetry Properties of the Fourier Transform2.9 Fourier Transform Theorems2.10 Discrete-Time Random Random Signals

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1497-Chapter2