Chapter 2. Discrete-Time Signals and Systems

Gao Xinbo

School of E.E., Xidian Univ.Xbgao@ieee.org

http://see.xidian.edu.cn/teach/matlabdsp/

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

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

)(txa

)(nx

}),1(),0(),1(,{)}({)( xxxnxnx

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?

}7,3,4,1,0,1,1,2{)( nx

];7,3,4,1,0,1,1,2[

];4,3,2,1,0,1,2,3[

x

n

Types of sequences Elementary sequence for analysis purposes 1. Unit sample sequence

Representation in MATLAB

,0,0,1,0,0,0,0

0,1)(

n

nn

201210

00 ,,

,0

,1)( nnnnnn

nn

nnnn

Function [x,n]=impseq(n0,n1,n2)

A: n=[n1:n2]; x = zeros(1,n2-n1+1); x(n0-n1+1)=1;

B: n=[n1:n2]; x = [(n-n0)==0]; stem(n,x,’ro’);

-3 -2 -1 0 1 2 30

0.2

0.4

0.6

0.8

1

n

(n

-n0)

-3<n<3

n0=0

2. Unit step sequence

,1,1,1,0,0,0,0

0,1)(

n

nnu

201210

00 ,,

,0

,1)( nnnnnn

nn

nnnnu

A: n=[n1:n2]; x=zeros(1,n2-n2+1); x(n0-n1+1:end)=1;

B: n=[n1:n2]; x=[(n-n0)>=0];

3. Real-valued exponential sequence

Rananx n ;,)(

For Example:

100,)9.0()( nnx n

n=[0:10]; x=(0.9).^n; stem(n,x,’ro’)

0 1 2 3 4 5 6 7 8 9 100

0.2

0.4

0.6

0.8

1

4. Complex-valued exponential sequence

nenx nj ,)( )( 0

Attenuation: 衰减因子frequency in radians:

For Example: n=[0:10]; x=exp((2+3j)*n);

5. Sinusoidal sequence

nnnx ),cos()( 0

Phase in radians

For Example:

n=[0:10]; x=3*cos(0.1*pi*n+pi/3)+2*sin(0.5*pi*n)

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]

7. Periodic sequence A sequence x(n) is periodic if x(n)=x(n+N) The smallest integer N is called the

fundamental period For example A: xtilde=[x,x,x,…x] B: xtilde=x’*ones(1,P); xtilde=xtilde(:);

xtilde=xtilde’; transposition

Operations on sequence 1. Signal addition

Sample-by-sample addition {x1(n)}+{x2(n)}={x1(n)+x2(n)}

Function [y,n]=sigadd(x1,n1,x2,n2)

n=min(min(n1),min(n2)): max(max(n1),max(n2));

y1=zeros(1,length(n)); y2=y1;

y1(find((n>=min(n1)) & (n<=max(n1))==1))=x1;

y2(find((n>=min(n2)) & (n<=max(n2))==1))=x2;

Y=y1 + y2;

2. Signal multiplication Sample-by-sample multiplication Dot multiplication {x1(n)}.{x2(n)}={x1(n) x2(n)}

Function [y,n]=sigmult(x1,n1,x2,n2)

n=min(min(n1),min(n2)) : max(max(n1),max(n2));

y1=zeros(1,length(n)); y2=y1;

y1(find((n>=min(n1)) & (n<=max(n1))==1))=x1;

y2(find((n>=min(n2)) & (n<=max(n2))==1))=x2;

Y=y1 .* y2;

3. Scaling a{x(n)}={ax(n)}

5. folding

y(n)={x(n-k)} m=n-k; y=x;

4. Shifting

y(n)={x(-n)} y=fliplr(x); n=-fliplr(n);

6. Sample summation

ss = sum(x(n1:n2);

7. Sample production

sp = prod(x(n1:n2));

2

1

)()()( 21

n

nn

nxnxnx

)()()( 21

2

1

nxnxnxn

nn

8. Signal energy

se = sum(x .* conj(x)); or

se = sum(abs(x) .^ 2);

9. Signal power

nnx nxnxnx 2* |)(|)()(

1

0

2|)(|1 N

nx nx

NP

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

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)

k

knkxnx )()()(

)]()([2

1)(

)]()([2

1)(

nxnxnx

nxnxnx

o

e

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]); m=m1:m2;

nm = n(1)-m(1); n1 = 1:length(n);

x1 = zeros(1, length(m)); x1(n1+nm) = x; x = x1;

xe = 0.5 * (x + flipflr(x));

xo = 0.5*(x - fliplr(x));

The geometric series A one-side exponential sequence of the form {an, n>=0},

where a is an arbitrary constant, is called a geometric series.

Expression for the sum of any finite number of terms of the series

1||,1

1

0

afor

aa

n

n

aa

aa

NN

n

n

,

1

11

0

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

nyx lnynxlr )()()(,

nxx lnxnxlr )()()(,

The index l is called the shift or lag parameter.

The special case: y(n)=x(n)

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

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)

)(),(,,

)]([)]([)]()([

2121

22112211

nxnxaa

nxLanxLanxanxaL

])([)()()()]([)(

nn

knLkxknkxLnxLny

)]([ knL

n

knhkxny ),()()(

h(n,k): the time-varying impulse response

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) = L[x(n)] --- y(n-k) = L[x(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).

)()]([),( knhknLknh

)(*)()()()]([)( nhnxknkkxnxLTInyk

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

n

nhStabilityBIBO |)(|

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 < 0 Such a sequence is termed a causal sequence.

To make sure that systems can be built.

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,ny]=conv_m(x,nx,h,nh) nyb = nx(1)+nh(1); nye = nx(length(x))+nh(length(h)); ny = [nyb:nye]; n = conv(x,h)

Function form of convolution

)()9.0(1)(*)()(

)()9.0()(),10()()(9

0

knunhnxny

nunhnununx

kn

k

n

Three different conditions under which u(n-k) can be evaluated:

Case 1: n<0 % the nonzero values of x(n)and y(n) do not overlap.

Case 2: 0<=n<9 % partially overlaps

Case 3: n>=9 % completely overlaps

Folded-and-shiftedSignals x=[x(1),x(2),x(3),x(4),x(5)]

System Impulse Response: h=[h(1),h(2)h(3),h(4)] y=conv(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.

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.

kkxx

kkyx

nxnxnkxnxknxnxnr

nxnynkxnyknxnynr

)(*)())(()()()()(

)(*)())(()()()()(

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.

nmnxbknyaM

mm

N

kk

,)()(

00

M

m

N

kkm knyamnxbny

0 1

)()()(

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

Zero-input and Zero-state response In DSP the difference equation is generally solved forward in

time from n=0. Therefore initial conditions on x(n) and y(n) are necessary to determine the output for n>=0.

Subject to the initial conditions:

0,)()()(0 1

nknyamnxbnyM

m

N

kkm

}1);({}1);({ nMnxandnNny

)()()( nynyny ZSZI Solution:

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

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 {a0=1} Implementation in Matlab: Conv(x,h); filter(b,1,x)

M

mm mnxbny

0

)()(

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

N

kk nxknya

0

)()(

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}

0,)()()(0 1

nknyamnxbnyM

m

N

kkm

Reference and Assignment Textbook: pp1 to pp35 Chinese reference book: pp1 to pp18

丁玉美、高西全编的“数字信号处理”（第二版），西电出版社 ,2001年 1月

Exercises: Textbook: p2.1b,c; p2.2b,d; 选 2.5 Textbook: P2.12b, 2.15, 2.17b, 选 2.8