# Lecture 4: Discrete-Time Systems - PowerPoint PPT Presentation

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n. n. Lecture 4: Discrete-Time Systems. Properties of Systems: linearity, time-invariance, causality, stability Properties of LTI Systems: impulse response finite impulse response (FIR) infinite impulse response (IIR) convolution. Processing Methods.

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Lecture 4: Discrete-Time Systems

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### Lecture 4: Discrete-Time Systems

• Properties of Systems:

• linearity, time-invariance, causality, stability

• Properties of LTI Systems:

• impulse response

• finite impulse response (FIR)

• infinite impulse response (IIR)

• convolution

EE421, Lecture 4

### Processing Methods

• Block processing: output values are computed by processing an entire block of input values.

• non real-time processing

• Sample processing: output values are computed by processing input samples one at a time.

• real-time processing

EE421, Lecture 4

### Processing Methods

• block processing

EE421, Lecture 4

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### Processing Methods

• sample processing

output

state variables

x(n)

y(n)

a0

a1

a2

a3

r

e

g

r

e

g

r

e

g

w1

w2

w3

EE421, Lecture 4

### FIR Filters

• Impulse response extends only over a finite time:

• filter order = M

• convolution equation:

filter coefficients, filter weights, filter taps

causal system

EE421, Lecture 4

### FIR Filters

• Examples

• y(n) = 2x(n) + 3x(n-1) + 5x(n-2) + 2x(n-4)impulse response = {. . ., 0, 0, 2, 3, 5, 0, 2, 0, 0, . . .}

• y(n) = (1/4)x(n+1) + (1/2)x(n) + (1/4)x(n-1)impulse response = {. . ., 0, 0, 1/4, 1/2, 1/4, 0, 0, . . .}

n=0

n=0

EE421, Lecture 4

### IIR Filters

• Impulse response extends over an infinite time:

• we cannot approach this in the same manner as an FIR filter

• IIR filters cannot, in general, be computed!

• we restrict our attention to systems described by difference equations:

y(n) = y(n-1) + 2y(n-2) + x(n) - x(n-1)

a0y(n) + a1y(n-1) + … + aMy(n-M) = b0x(n) + b1x(n-1) + … + bLx(n-L)

EE421, Lecture 4

### IIR Filters

• Examples:

• y(n) = ay(n-1) + x(n)

• impulse response: h(n) = ah(n-1) + d(n)

• h(n) = anu(n)

• y(n) = ay(n-2) + x(n)

• impulse response: h(n) = ah(n-2) + d(n)

• h(n) = an/2, if n is even; h(n) = 0 otherwise

EE421, Lecture 4

### IIR Filters

• Examples:

• y(n) = ay(n-1) + x(n) + x(n-1)

• impulse response: h(n) = ah(n-1) + d(n) + d(n-1)

• h(n) = 1 if n=0, h(n) = an-1(1+a)u(n) otherwise

EE421, Lecture 4