INTRODUCTION TO FILTERS

KEYPOINTS

• What is a filter?• What are preliminary conditions to know to

design a filter?• What does a simple analog filter do?• What does a simple digital filter do?• Impulse response? Why impulse response?• Convolution? What is this?• Filters – FIR VS IIR

FILTERS

• Filters may be classified as either digital or analog:

• Digital filters are implemented using a digital computer or special purpose digital hardware.

• Analog filters may be classified as either passive or active and are usually implemented with R, L and C components and operational amplifiers.

The Time and Frequency Response of the Analog Filter

Time (impulse) Response

Frequency Response

what’s the “impulse response”?Impulse response is the response of the circuit to a

(mathematical) signal of infinite height (and power), and infinitely short duration. This “unit impulse” has very special characteristics:

1) It contains all frequencies2) It has energy of “1” at all frequencies.3) It describes the behavior of the filter at all

frequencies. Completely.

POINT 3 is a key point to remember. The impulse response of a filter is a complete description of what it does.

Another way to look at the impulse response.

• The impulse response of a system shows how a filter captures the HISTORY of the signal. In other words: – The value of the impulse response at a time ‘t’

demonstrates how much of the HISTORY of the signal is added to the output at time ‘t’ later.

How about a “Digital” filter?

+

One SampleDelay

Feedback

X

Input Output

The digital filter uses history explicitly.

What is it’s time and frequency response?

The result is the same (below half the sampling rate)

Convolution• The process of multiplying the time-reversed

signal by the impulse response, and summing (integrating) is called “convolution”.

t

tthtTx )()(

EXPRESSING A TRANSFER FUNCTION

S() H() Y() = S()*H()

The usual way we see transfer functions expressed:(notice this is in the ‘’, or frequency, domain)

What’s actually happening in the time domain:(Note: is used here to denote convolution. There are other notations.)

s(t) s(t)h(t) y(t)

• Multiplication in the time domain is the same as convolution in the frequency domain.

• Multiplication in the FREQUENCY domain is the same as convolution in the TIME domain.

• It works either way. This is duality in action.• For typical filters, convolution is what happens in the

time domain.• Convolution is merely another way of expressing

what happens when you filter a signal.• It’s the same as multiplying the signal by the transfer

function.

There are other tradeoffs possible• IIR filters can have:

– Passband ripple only– Stop band ripple only– Neither passband nor stop band ripple (monotonic

response)– Both passband and stop band ripple

• FIR filters as usually designed can have:– Ratio of passband ripple to stop band ripple

controlled via design parameters. The filter response is not defined in a “transition” band.

– There are other FIR types possible, they are not that common in most present-day uses.

So, they are the same?• Well, no. In fact, FIR filters have zeros, and IIR filters

have poles. (In reality, nearly all IIR filters have both poles and zeros, which is to say that they have both an FIR and an IIR part.

• FIR and IIR filters can have quite different properties, and usually do, they are two different means to an end. Neither one nor the other is always better.

How are filters described?• FIR filters usually are simply listed by either the tap

weights (individual values) or by a function that describes the tap weights.

• IIR filters are described as sets of poles and zeros.

Poles? Zeros? WHAT!?• Poles and zeros are a way of expressing a transfer

function as two polynomials, one in the numerator, and one in the denominator.

• For either numerator or denominator, a polynomial can be described as a1+ a2 * z1 + a3* z2 … where a1, a2, a3 are the “tap weights’.

• One can also calculate the roots of the polynomial.– The roots of the numerator are the ZEROS.– The roots of the denominator are the POLES.

Why poles? Why Zeros?• A zero shows a value for the polynomial variable that

results in a ZERO output.• A pole shows a value for the polynomial that has an

INFINTE output. (the response looks like a pole)

• The meaning of poles and zeros in terms of frequency changes depending on the kind of transfer function (i.e. ‘s’ or Laplace domain, ‘z’ domain, ‘w’ or Fourier domain, or others) but for the commonly used domains will still be some expression of frequency.

A pole/zero plot for a 5th order Butterworth

STOP BAND• A stopband is a band of frequencies, between

specified limits, through which a circuit, such as filter or telephone circuit, does not allow signals to pass, or the attenuation is above the required stopband attenuation level.

• Depending on application, the required attenuation within the stopband may typically be a value between 20 and 120 dB higher than the nominal passband attenuation, which often is 0 dB.

PASS BAND AND ROLL OFF• A passband is the range of frequencies or

wavelengths that can pass through a filter without being attenuated.

• Roll-off is a term commonly used to describe the steepness of a transmission function with frequency, particularly in electrical network analysis, and most especially in connection with filter circuits in the transition between a passband and a stopband.

• The most popular filters include Butterworth, Chebyshev and elliptic methods.