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Lecture 4: Discrete-Time Systems

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n

n

- Properties of Systems:
- linearity, time-invariance, causality, stability

- Properties of LTI Systems:
- impulse response
- finite impulse response (FIR)
- infinite impulse response (IIR)

- convolution

- impulse response

EE421, Lecture 4

- 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

- block processing

EE421, Lecture 4

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+

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- 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

- Impulse response extends only over a finite time:
- filter order = M
- convolution equation:

filter coefficients, filter weights, filter taps

causal system

EE421, Lecture 4

- 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

- 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

- 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

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

EE421, Lecture 4

- 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