1. 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 n n
22-Sept, 98
EE421, Lecture 4

2. 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
22-Sept, 98
EE421, Lecture 4

3. Processing Methods
block processing
22-Sept, 98
EE421, Lecture 4

4. Processing Methods sample processing output state variables + + + r e
x(n) + + +
y(n) a0 a1 a2 a3 r e g r e g r e g w1 w2 w3
22-Sept, 98
EE421, Lecture 4

5. FIR Filters Impulse response extends only over a finite time:
filter order = M
convolution equation:
filter coefficients, filter weights, filter taps
causal system
22-Sept, 98
EE421, Lecture 4

6. 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
22-Sept, 98
EE421, Lecture 4

7. 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)
22-Sept, 98
EE421, Lecture 4

8. 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
22-Sept, 98
EE421, Lecture 4

9. 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
22-Sept, 98
EE421, Lecture 4