Approximation of Derivatives

Spring 2012

Introduction

IIR filter design techniques depend mainly on converting analogue filter to a digital one.

Analogue filter can be described by three methods 1) Its system function. M   k sk k 0 H a (s)  N  k sk k 0 2) By its impulse response which is related to the system function by

 st H a (s)  ha (t)e dt 

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١ Introduction

3) By linear constant-coefficient differential equation d k y(t) M d k x(t)     k k  k k dt k 0 dt

Each of these three characterizations leads to alternative method for converting the analog filter to its digital equivalents

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Introduction

The system function Ha(s) is stable if all its poles lie in the left half of the s-plane. Hence, the conversion is effective if

1. The j axis in the s-plane map into the unit circle in the z-plane. 2. The left-half plane LHP of the s-plane map into the inside of a unit circle in the z-plane (i.e stable analogue filter will be converted to stable digital filter.

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٢ Approximation of derivatives

The simplest methods to convert an analogue filter to a digital filter is to convert the differential equation into equivalent difference equation.

For the first derivative dy(t) y(nT)  y(nT  T) y(n)  y(n 1)   dt tnT T T

y(t) H (s)  s dy(t) dt

1 z 1 y(n) H (z)  y(n)  y(n 1) T T

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Approximation of derivatives

For the analog derivative, the system function is given by H (s)  s

Whereas, the digital system function is H (z)  (1 z 1 ) /T Which leads to 1 z 1 s  T

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The 2nd derivative is replaced by the second difference

d 2 y(t) d dy(t) [y(nT )  y(nt T)]/ T [y(nT T )  y(nT  2T)]/T   2   dt dt  dt t nT T y(n)  2y(n 1)  y(n  2)  T 2 In frequency domain this is equivalent to

2 1 2z 1  z 2 1 z 1  s 2     2   T  T 

Then the kth derivative is equivalent frequency-domain relationship

1 k k 1 z  s     T  ٧ Digital Signal Processing ٧ Slide

Approximation of derivatives

The system function for IIR filter obtained as a result of approximation of the derivatives is H (z)  H (s) a s(1z 1 ) /T Let us investigate the implication of the mapping from the s-plane to the z-plane 1 z  1  sT Then the j  axis is mapped into 1 z  1 jT 1 T   j 1  2T 2 1  2T 2

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٤ Approximation of derivatives

The y -axis in the s -plane is mapped into a circle of radius ½ and with centre z = ½. Any point in the left hand plane will be mapped to points inside that circle. So the resultant filter is stable

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Approximation of derivatives

The possible locations of poles of the digital filter are confined to small frequencies.

As consequence, the mapping is restricted to low-pass and band-pass filters with relatively small resonant frequencies.

It is not possible to transform a high-pass analog filter into a corresponding high-pass digital filter.

In attempt to overcome this limitations, more complex substitutions for the derivatives have been proposed such as

dy(t) 1 L y(nT  kT)  y(nT  kT) tnT   k dt T k 1 T

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The resulting mapping between the s-plane and the z-plane

is now L 1 k k s   k z  z  T k0

When z = ej j2 L s   k sink  j T k1

Which is pure imaginary, which means by carefully selecting ks, The j axis can be transformed into the unity circle. This can be done by using optimization

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Approximation of derivatives

Example: Convert the analogue bandpass filter with system function 1 H s  a (s  0.1)2  9 Into a digital IIR filter by substituting for the derivatives method 1 H (z)  ((1 z 1 ) / T  0.1) 2  9

T 2 /(1 0.2T  9.01T 2 ) H (z)  2(1 0.1T ) 1 1 z 1  z 2 1 0.2T  9.01T 2 1 0.2T  9.01T 2

T should be less than or equal 0.1 for poles near the unity circle

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٦ Approximation of derivatives

Example: Convert the analogue bandpass filter in the previous example by use of the mapping 1 s  z  z 1  T Solution: by substituting for s in H(s), we obtain

1 H (z)  ((z  z 1 ) /T  0.1) 2  9

z 2T 2 H (z)  z 4  0.2Tz 3  (2  9.01T 2 )z 2  0.2Tz 1

H (z) has four poles while H (s) has two poles which means that the conversion has led to more complex system

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