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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/

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

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

Types of sequences

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

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)

2. Unit step sequence

A nn1n2 xzeros(1,n2-n21)

x(n0-n11end)1 B nn1n2 x(n-n0)gt0

3. Real-valued exponential sequence

For Example

n010 x(0.9).n stem(n,x,ro)

4. Complex-valued exponential sequence

Attenuation frequency in radians For

Example n010 xexp((23j)n)

5. Sinusoidal sequence

Phase in radians For Example n010

x3cos(0.1pinpi/3)2sin(0.5pin)

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(nN)
- The smallest integer N is called the fundamental

period - For example
- A xtildex,x,x,x
- B xtildexones(1,P) xtildextilde()

xtildextilde transposition

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

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

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)

6. Sample summation ss

sum(x(n1n2) 7. Sample production sp

prod(x(n1n2))

8. Signal energy se sum(x . conj(x))

or se sum(abs(x) . 2) 9. Signal power

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)

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))

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

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)

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)

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) 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).

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

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.

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)

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

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.

(No Transcript)

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.

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.

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 n0. Therefore

initial conditions on x(n) and y(n) are necessary

to determine the output for ngt0. - Subject to the initial conditions

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 a01
- Implementation in Matlab Conv(x,h) filter(b,1,x)

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

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

Reference and Assignment

- Textbook pp1 to pp35
- Chinese reference book pp1 to pp18
- (),,20011
- Exercises
- Textbook p2.1b,c p2.2b,d 2.5
- Textbook P2.12b, 2.15, 2.17b, 2.8

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