Iir filters
This presentation is the property of its rightful owner.
Sponsored Links
1 / 46

IIR Filters PowerPoint PPT Presentation


  • 89 Views
  • Uploaded on
  • Presentation posted in: General

IIR Filters. The general form for z -transform transfer functions for a digital filter is. The inverse of this z -transform transfer function h[n] is the impulse response for this digital filter. As an example if. we have.

Download Presentation

IIR Filters

An Image/Link below is provided (as is) to download presentation

Download Policy: Content on the Website is provided to you AS IS for your information and personal use and may not be sold / licensed / shared on other websites without getting consent from its author.While downloading, if for some reason you are not able to download a presentation, the publisher may have deleted the file from their server.


- - - - - - - - - - - - - - - - - - - - - - - - - - E N D - - - - - - - - - - - - - - - - - - - - - - - - - -

Presentation Transcript


Iir filters

IIR Filters


Iir filters

The general form for z-transform transfer functions for a digital filter is

The inverse of this z-transform transfer function h[n] is the impulse responsefor this digital filter.


Iir filters

As an example if

we have


Iir filters

The inverse z-transform of this transfer function (the impulse response) is

h[n]

n

1

2

3

4

5


Iir filters

This impulse response function goes on forever. That is, the impulse response is infinite in duration. Such filters are referred to as infinite impulse response or IIR filters.


Iir filters

Another example would be

The corresponding impulse response is


Iir filters

h[n]

n

1

2

3

4

5

This impulse response is finite in duration. Such filters are referred to as finite impulse response or FIR filters.


Iir filters

Given a transfer function, how do we know whether this transfer function corresponds to an impulse response that is infinite or finite in duration?

In our previous two examples we had

(IIR)

(FIR)


Iir filters

In general, any filter whose transfer function has a denominator (that does not factor-out) will have an impulse response that is infinite in duration corresponding to an IIR filter.

Any filter whose transfer function does not have a denominator will have an impulse response that is finite in duration corresponding to an FIR filter.


Iir filters

The distinction between IIR filter transfer functions and FIR filter transfer functions becomes more clear if we look at the corresponding difference equations. In our examples the corresponding difference equations are

(IIR)

(FIR)


Iir filters

The IIR filter difference equation is recursive in nature: the current output depends upon the previous output

Since the current output depends upon the previous output and the previous output depends upon its previous output, the output depends upon the infinite past.


Iir filters

The FIR filter difference equation is depends only upon the input

If the input is finite in duration (such as an impulse) then the output is finite in duration. The output depends upon the finite past.


Iir filters

The difference between IIR and FIR filters can also be seen by looking at the transfer functions and noting that the IIR transfer function can be expanded using a geometric series that is infinite:

The FIR filter is already in a finite series form.


Iir filters

Most analog filters have an impulse response which is infinite in duration. IIR filters are generally designed by emulating an analog prototype filter. There are two methods for doing this analog filter emulation:

(1) the matched z-transform or impulse invariant

transform

(2)the bilinear transformation.

In both cases, we are given an analog transfer function H(s), and we transform this function into a digital transfer function H(z).


The matched z transform

The Matched z-Transform

In the matched z-transform digital filter design method we try to “match” the impulse response of the analog filter with that of the digital filter being designed.

To match the impulse responses, we take the inverse Laplace transform of the analog filter H(s)h(t), then sample the impulse response h(t)h[n], then take the z-transform of the sampled impulse response to get the z-transform transfer function h[n]H(z).


Iir filters

Analog Prototype

Digital Filter

L-1

Z

sample

Once we have our z-transform transfer function H(z), we apply the definition of the transfer function to write our digital filter equations:


Iir filters

Example: Use the matched filter design method to design the digital equivalent of an integrator.

Solution: The analog transfer function is

The inverse Laplace transform is


Iir filters

We then sample the impulse response to get h[n]:

Finally, we take the z-transform of the impulse response to get the digital filter transfer function.


Iir filters

Finally, we apply the definition of the z-transform transfer function to get the relationship between the input of the digital filter x[n] and the output of the digital filter y[n].


Iir filters

Applying this formula to an arbitrary input x[n], we have


Iir filters

We see that the output is the summationof the input. Thus the digital filter accurately represents the analog filter.

The digital filter does not always accurately represent the analog filter as will be seen in the next example.


Iir filters

Example: Use the matched filter design method to design the following transfer function:

where Wc = Ws/4, and Ws is the sampling frequency.

Solution: First, we find the impulse response


Iir filters

Then we sample the impulse response:

Then we take the z-transform of the (discrete-time) impulse response:


Iir filters

We then find the frequency response of the filter


Iir filters

As we can see, it is not much of a low-pass filter: the frequency rolloff is not very great. The reason for this small rolloff is aliasing error: the frequency response of H(z) is composed of copies of the frequency response of H(s) at 0, Ws, 2Ws, etc. The copy at 2Ws overlaps the copy at 0.

Aliasing error is an inherent problem in matched filter design. Unless the cutoff frequency is very low compared to the sampling frequency, we will get substantial error due to aliasing. This aliasing problem is solved using the bilinear transformation.


The bilinear transformation

The Bilinear Transformation

The bilinear transformation is a fairly direct method of converting H(s) to H(z). Rather than map the analog frequencies W=0 to Ws/2 to the digital frequencies w=0 to w=p (as we had done with the matched z-transform), we will map the analog frequencies W=0 to to the digital frequencies w=0 to w=p

Matched z-transform: W=(0,Ws/2)  w=(0,p)

Bilinear Transformation: W=(0,)  w=(0,p)


Iir filters

What kind of function maps W=(0,)  w=(0,p)?

How about this:

As we can see from the graph on the following slide, this function does perform the necessary mapping.


Iir filters

We know the relationship between W and w; what is the relationship between s and z?


Iir filters

Since s=jW and z=ejw, we have


Iir filters

Because of the “warped” nature of our transformation, it is necessary to “pre-warp” our analog prototype critical frequencies so as to coincide with the critical frequencies of the corresponding digital filter.


Iir filters

Example: Use the bilinear transformation method to find the digital equivalent to the following transfer function:

where Wc = Ws/4, and Ws is the sampling frequency.

Solution: First, we must “pre-warp” the analog frequency:


Iir filters

We then substitute our “pre-warped” frequency,

and apply the bilinear (sz) transformation:


Iir filters

The frequency response of the filter is


Iir filters

As can be seen, the frequency response is much improved. At w=0, the response is the same as the analog filter at W=0, and at w=p, the frequency response is the same as the analog filter at W=.


Iir filters

Example: Repeat the previous example

where Wc = Ws/6.

Solution: Our “prewarping” is slightly different:


Iir filters

Substituting our “pre-warped” frequency, we have


Iir filters

Applying the bilinear transformation, we have

The frequency plot is given on the following slide.


Iir filters

Example: find the digital equivalent of a second-order Butterworth filter using the bilinear transformation. Let Wc = Ws/4.

Solution: The second-order Butterworth filter has the following form:

(Enter [b a] = butter(2,1,'s') in MATLAB.)


Iir filters

The prewarping is the same as in the first example:

So,


Iir filters

Applying the bilinear transformation we have


Iir filters

The frequency response is on the following slide.


  • Login