Lec5: The IIR filter

Question 1

what is the IIR filter?

Answer 1

Infinite Impulse Responce (IIR)

Question 2

The General IIR filter equation?

Answer 2

Question 3

a_l and b_k ?

Answer 3

The set of coefficients that describe the filter,

Question 4

How is the FIR different to the IIR?

Answer 4

It is similar to the FIR filter (the feedforward term), but now we have introduced a feedback term. I.E the output from the filter is passed back into the system.

Question 5

The total coefficient count of IIR?

Answer 5

M+N+1. The filter order is governed by N

Question 6

The assumptions of the IIR filter?

Answer 6

For the IIR filter, we make the assumption of initial rest.

Formally: x[n] = 0 for n < n₀ and y[n] = 0 for n < n₀. Where n₀ is our start time.

This is sufficient to result in the IIR filter being both linear and time-invariant.

Question 7

Why are the initial filter assumptions required?

Answer 7

This is sufficient to result in the IIR filter being both linear and time invariant.

The principle of superposition also holds, because the difference equation involves only linear combinations of the inputs and outputs.

Question 8

How to demonstrate a basic IIR filter?

Answer 8

We can demonstrate the basic IIR filter by considering the impulse response ℎ[n] for a 1st order filter:

Question 9

Find the impulse response?

Answer 9

Question 10

Impulse response graph for a first-order IIR with b₀=1 and a₁ = 0.8

Answer 10

Question 11

The z transform of both the feedforward and feedback parts?

Answer 11

Question 12

Finding the IIR system function?

Answer 12

Question 13

The numerator and denominator polynomials?

Answer 13

The coefficients of the numerator polynomial B(z) correspond to the feed-forward terms.

Those of the denominator polynomial A(z) for z^-l where l>0 correspond to the feed- back terms.

Question 14

Finding the poles and zeros of the system function H(z)?

Answer 14

Question 15

Calculating the poles and the zeros?

Answer 15

Question 16

IIR filter, how to examine the frequency response?

Answer 16

Question 17

Filter details

Answer 17

Highpass

Note: At DC, the ratio is not quite 0 but 0.55

Question 18

How to conduct frequency response on a filter?

Answer 18

Question 19

Filter type?

Answer 19

Bandpass

Question 20

How can IIR be unstable?

Answer 20

Due to the feedback path (and the infinite impulse response), we have the potential for instability in IIR filters. If the feedback is self sustaining.

Question 21

The stability of causal IIR systems?

Answer 21

Causal IIR systems, initially at rest, are stable if all poles lie inside the unit circle.

Question 22

The stability of causal FIR systems?

Answer 22

Causal FIR systems, which have no poles (other than at zero) are unconditionally stable.

Question 23

How can the stability of a 1st order IIR be investigated?

Answer 23

Question 24

Answer 24