DSP-Lec 03-Discrete Time Systems

Discrete-time signal™ The discrete-time signal xn is obtained from sampling an analog signal t i e n= nT here T is the sampling period signal xt, i.e., xn=xnT where T is the sampling per

™ I t/ t t l ti hi f th t

™ Input/output relationship of the systems

™ Linear time-invariant (LTI) systems

™ FIR d IIR fil

1 Discrete-time signal

™ The discrete-time signal x(n) is obtained from sampling an analog

signal (t) i e (n)= (nT) here T is the sampling period

signal x(t), i.e., x(n)=x(nT) where T is the sampling period

™ There are some representations of the discrete-time signal x(n):

Some elementary of discrete-time signals

™ Unit sample sequence (unit impulse):

( )

for n n

2 Input/output rules

™ A discrete-time system is a processor that transform an input

seq ence (n) into an o tp t seq ence (n)

sequence x(n) into an output sequence y(n)

™ Sample by sample processing:

Fig: Discrete-time system

™ Sample-by-sample processing:

that is, and so on

™ Block processing:

Basic building blocks of DSP systems

) ( )

( )

™ Signal multiplier x1( n ) y ( n ) = x1( n ) x2( n )

™ Let x(n)={1, 3, 2, 5} Find the output and plot the graph for the ( ) { , , , } p p g p

systems with input/out rules as follows:

a) y(n)=2x(n)) y( ) ( )

b) y(n)=x(n-4)

c) y(n)=x(n)+x(n 1)

™ A weighted average system y(n)=2x(n)+4x(n-1)+5x(n-2) Given the g g y y( ) ( ) ( ) ( )input signal x(n)=[x0,x1, x2, x4 ]

a) Find the output y(n) by sample-sample processing method?) p y( ) y p p p g

b) Find the output y(n) by block processing method

c) Plot the block diagram to implement this system from basic

c) Plot the block diagram to implement this system from basic

building blocks ?

3 Linearity and time invariance

™ A linear system has the property that the output signal due to a

linear combination of t o inp t signals can be obtained b forming

linear combination of two input signals can be obtained by forming the same linear combination of the individual outputs

Fig: Testing linearity

™ If y(n)=a1y1(n)+a2y2(n) ∀ a1, a2 Æ linear system Otherwise, the

system is nonlinear

3 Linearity and time invariance

™ A time-invariant system is a system that its input-output

characteristics do not change ith time

characteristics do not change with time

Fig: Testing time invarianceg g

™ If yD(n)=y(n-D) ∀ DÆ time-invariant system Otherwise, the

system is time-variant

4 Impulse response

™ Linear time-invariant (LTI) systems are characterized uniquely by their impulse response sequence h(n), which is defined as the

response of the systems to a unit impulse δ(n)

Fig: Impulse response of an LTI system

5 Convolution of LTI systems

Fig: Response to linear combination of inputs

™ Convolution:

(LTI form)

) ( )

( )

( ) ( )

( )

( ) ( )

5 FIR and IIR filters

™ A finite impulse response (FIR) filter has impulse response h(n) that extend only over a finite time interval say 0 ≤n ≤ M

that extend only over a finite time interval, say 0 ≤n ≤ M

Fi FIR i lFig: FIR impulse response

™ M: filter order; Lhh=M+1: the length of impulse response g p p

™ h={h0, h1, …, hM} is referred by various name such as filter

coefficients, filter weights, or filter taps

™ The third-order FIR filter has the impulse response h=[1, 2, 1, -1]

a) Find the I/O equation, i.e., the relationship of the input x(n) and the output y(n) ?

b) Given x=[1, 2, 3, 1], find the output y(n) ?

5 FIR and IIR filters

™ A infinite impulse response (IIR) filter has impulse response h(n)

of infinite duration say 0 ≤n ≤ ∞

of infinite duration, say 0 ≤n ≤ ∞

Fi IIR i lFig: IIR impulse response

) ( )

( )

( )

(

m

m n

x m h n

x n

h n

y

™ IIR filtering equation:

™ The I/O equation of IIR filters are expressed as the recursive

difference equation

n m

n

m k

5 0 ( 4

)

n for

n

a) Find the I/O difference equation ?

b) Find the difference equation for h(n)?

b) Find the difference equation for h(n)?

6 Causality and Stability

Fig: Causal, anticausal, and mixed signals

™ LTI systems can also classified in terms of causality depending on whether h(n) is casual, anticausal or mixed

™ A system is stable (BIBO) if bounded inputs (|x(n)| ≤A) always generate bounded outputs (|y(n)| ≤B).

™ A LTI system is stable ⇔ ∑∞ < ∞

™ Consider the causality and stability of the following systems:

a) h(n)=(0.5)nu(n)

b) h(n)=-(0.5)) ( ) ( ) (nu(-n-1))

™ Problems: 3.1, 3.2, 3.3, 3.4, 3.5, 3.6