Finite Impulse Response (FIR) and Infinite Impulse Response (IIR) Syllabus:Finite impulse Response (FIR) Filters:Introduction, magnitude response and phase response of digital filters, frequency response of linear phase FIR filters, Design techniques of FIR filters, design of optimal linear FIR filters.Infinite Impulse Response (IIR) Filters :Introduction, IIR filter design by approximation of derivatives, IIR filter design by impulse invariant method, nr filter design by the bilinear transformation, Butterworth filters, Chebyshev filters, Elliptic filters, frequency transformation.ContentsPage No.7.1 Concept of Filtering7-27.2 Design Techniques FIR Filter7-37.3 Design of Optimal Linear Phase FIR Filters7-47.4 Properties of Commonly used Windows7-57.5 Solved Problems on Windowing method7-117.6 Design of IIR Filters using Approximation of Derivatives7-167.7 Impulse Invariant Method7-187.8 Bi-linear Transform Filter (BLT)7-297.9 Comparison between Impulse Invariance and Bilinear Transformation Method7-317.10 Solved Problems using BLT7-317.11 Basic Filter Approximations7-387.12 Frequency Transformations7-437.13 Comparison between IIR and FIR Filters7-48

introduction :The basic function of digital filter is to, eliminate the noise and to extract the signal of interlay from other signals. A digital filter is a basic device used in digital signal processing.There are several techniques available to design the digital filters. But generally while desig a digital filter; first an analog filter is designed and then it is converted into the corresponding dir._ filter.Before studying digital filters, we will study some fundamental things related to analog filtering. This is necessary because, generally digital filters are designed using analog filters.Some parameters related to analog filters :1. Pass band : It passes certain range of frequencies. In the pass band, attenuation is zero.2. Stop band : It suppresses certain range of frequencies. In the stop band, attenuation is infantry.3. Cut-off frequency : This is the frequency which seperates pass band and stop band.7.1 Concept of Filtering :Analog filters are designed using analog components like resistors (R), inductors (L) and i capacitors (C). While digital filters are implemented using difference equation.The digital filters described by differential equations can be implemented using software like C or assembly language. We can easily change the algorithm; so we can easily change the filter characteristics according to our requirement.Basically there are two types of filters as follows :1. FIR (Finite impulse response) filter.2. IIR (Infinite impulse response) filter.We will study each type in detail; later in this chapter. Presently we will compare analog arc digital filters by studying advantages and disadvantages of digital filters.7.1.1 Advantages of Digital Filters :1. Many input signals can be filtered by one digital filter without replacing the hardware.2. Digital filters have characteristic like linear phase response. Such characteristic is not possible s obtain in case of analog filters.3. The performance of digital filters, does not vary with environmental parameters. But re environmental parameters like temperature, humidity etc., change the values of components :: case of analog filters. So it is required to calibrate analog filters periodically.4. In case of digital filters; since the filtering is done with the help of digital computer, both filtered and unfiltered data can be saved for further use.5. Unlike analog filters; the digital filters are portable.6. From unit to unit the performance of digital filters is repeatable.7. The digital filters are highly flexible.8. Using VLSI technology; the hardware of digital filters can be reduced. Similarly the parry consumption can be reduced.9. Digital filters can be used at very low frequencies, for example in Biomedical applications.

10. In case of analog filters; maintenance is frequently required. But for digital filters it is not required.7.1.2 Disadvantages of Digital Filters :1.Speed limitation :In case of digital filters, ADC and DAC are used. So the speed of digital filter depends on the conversion time of ADC and the settling time of DAC. Similarly the speed of operation of digital filter depends on the speed of digital processor. Thus the bandwidth of input signal processed is limited by ADC and DAC. In real time applications, the bandwidth of digital filter is much lower than analog filters.2.Finite wordlength effect:The accuracy of digital filter depends on the wordlength used to encode them in binary form. Wordlength should be long enough to obtain the required accuracy.The digital filters are also affected by the ADC noise, resulting from the quantization of continuous signals. Similarly the accuracy of digital filters is also affected by the round off noise occurred during computation.3.Long design and development time :An initial design and development time for digital hardware is more than analog filters.7.2 Design Techniques FIR Filter : FIR stands for Finite Impulse Response. FIR filters are called as non-recursive filters because they do not use the feedback. Before studying the design of FIR filters; we will discuss one important characteristic of FIR filter.7.2.1 FIR Filters are Inherently Stable :We know that LSI system is said to be stable if bounded input produces bounded output (BIBO).

We have the difference equation of FIR filter,

...(1.2.4)

Expanding Equation (7.2.1) we get,y(n) = b0x(n) + b1x(n-l) + .... + bM_1x(n-M+ 1)...(7.2.5Using Equation (7.2.4) we get,y(n) = h(0)x(n) + h(l)x(n-l) ++ h(M-1 )x(h-M +l)...(7.2.6Here h (0 ), h ( 1)... are constants that means they are bounded. Now from Equation (7.2.6); the output will be bounded if we apply bounded input. That means for every bounded input; the output of FIR filter is bounded. Thus FIR filters are inherently stable.7.2.2 FIR Filters are Linear Phase Filters (Magnitude and Phase Response):We will discuss the symmetry and antisymmetry of FIR filters. These conditions are related k their unit sample response h (n).The unit sample response of FIR filter is symmetric if it satisfies the condition.h(n) = h(M-l-n)n = 0,lM-1...(7.2.7Here M = Number of samples; so if M = 8 we get, Forn = 0 =>h(0) = h( 8- l -0) = h (7) For n = 1 => h (1 ) = h ( 8 - 1 - 1) = h ( 6) etc. If h ( n) is symmetric then, the filter is symmetric. Now unit sample response of FIR filter is antisymmetric if it satisfies the condition,h(n) = -h(M-l-n), n = 0,l....M-l...(7.2.8If this condition is satisfied then the filter is antisymmetricNow the phase of FIR filter is given by,

This equation shows that the phase of FIR filter is piecewise linear. Thus for the symmetric and antisymmetric FIR filters; the condition for linear phase is,h(n) = ;J)7.3 Design of Optimal Linear Phase FIR Filters :Different types of windows are used to design FIR filter. First we will discuss the design of FIR filter using rectangular window. The rectangular window is as shown in Fig. 7.3.1(a).

It is denoted by WR (n). Its magnitude is 1 for the range, n = 0 to M - 1. Now let hd ( n ) be the impulse response having infinite duration. If hd (n) is multiplied by WR (n) then a finite impulse response is obtained as shown in Fig. 7.3.1(b). That means we will get only limited pulse of hd (n); notall (oo) pulse. Since we are truncating the input sequence by using a window, this process is called as truncation process. Since the shape of window function is rectangular; it is called as rectangular window.7.4 Properties of Commonly used Windows :Some other types of window runcuons are as follows :(1) Hamming window(2) Hanning window(3) Triangular (Bartlett) window (4) Blackman window (5) Kaiser windowWe will consider that the range of each window is from - Q to Q. Here Q is positive integer number. (1) Hamming window :Hamming window function is pvea by.

Fig. 7.4.1 shows shape of Hamming window.

- *-Wh (n) is a bell-shaped sequence that is symmetric about n = 0. (2) Hanning window function :

Hanning window function is also called as Raised-cosine window. The function is denoted by,

Keter rigs. /.4.1(a), (b) and (c) to have a look to the spectrum.Magnitude of first side-lobe level is - 31 dB relative to maximum value.(3) Triangular window function OR Bartlett window :Triangular function is like tapering the rectangular window sequence linearly from the middl to the ends. Triangular window function can be given by

Window and its spectrum is shown in Fig. 7.4.2.Sidelobe level is smaller than that of rectangular window.Triangular window produces smoother magnitude response than that of rectangular window function.The transition from passband to stopband is not as steep as that for the rectangular window. In the stopband, the response is smoother, but attenuation is less than that produced by rectangular window, therefore because of this characteristic, triangular window is not usually good choice.

If we compare hanning window with triangular then hanning window function is smoother at the ends. Smoother ends reduces sidelobe level, while broaden middle section.4) Blackman window function :In blackman window function, we will find one more additional term in comparision with amming and hanning window. Because of additional cosine term, sidelobs are reduced further.

Window and its spectrum are shown in Fig. 7.4.4.Window function for blackman can be given by ,

Fig. 7.4.4 : Blackmail window function(5) Kaiser window function :Kaiser window function can be defined as,

Where I0 (x) is modified bessel function of the first kind and zero order. The tradeoff berw* main lobe width and side lobe level can be adjusted by varying parameter a.Here a is independent variable.6 can be expressed as

Table 7.4.1 shows the summary of window functions.Table 7.4.1

7.4.1 Gibb's Phenomenon:The impulse response of FIR filter in terms of rectangular window is given by,h(n) =hd(n)-WR(n)...(7.4.1)The frequency response of filter is obtained by taking Fourier transform of Equation (7.4.1). .-. H(co) = FT{hd(n)-WR(n)}.-. H((0) = Hd(co)*WR(o)...(7.4.2)This shows that the frequency response of FIR filter is equal to the convolution of desired frequency response, Hd ( go) and the Fourier transform of window function.

Now the desired frequency response of low pass FIR is shown in Fig. 7.4.1(a); while the frequency response of FIR filter obtained because of windowing is shown in Fig. 7.4.1(b).The sidelobes are present in the frequency response of window function. Because of the-: sidelobes; the ringing is observed in the frequency response of FIR filter. This ringing is predominant present near the bandedge and it is known as Gibb's phenomenon.Now the question arises, why the side lobes are present in the frequency response of wind

function ? This is because of the sudden discontinuities in the window function. Observe the magnitude response of rectangular window. In this case, the discontinuity is very abrupt. Therefore the sidelobes are of larger amplitude. Thus the ringing effect is maximum in case of rectangular window.Because of this reason; other window functions are developed which will not have the abrupt discontinuities. That means the window function will change more gradually in the time domain.7.4.2 Advantages and Disadvantages of Window Method :Advantages:1. The windowing method requires minimum amount of computational effort; so window method m simple to implement.2. For the given window; the maximum amplitude of ripple in the filter response is fixed. Thus e stopband attenuation is fixed in the given window.Disadvantages:1. The designing of FIR filters using windows is not flexible.2. The frequency response of FIR filter shows the convolution of spectrum of window function and desired frequency response. Because of this; the pass band and stop band edge frequencies caroche* be precisely specified.3. In many applications the expression for the desired filter response will be too complicated.Design steps for FIR filter:1. Get the desired frequency response, Hid (co).

2. Take inverse Fourier transform (IFT) of Hd (co) to obtain hd (n) - 3. Decide the length of FIR filter. 4. Multiply hd (n) by selected window function to get h (n).5. From h (n ) obtain H (Z ) and then realize it, if asked.7.5 Solved Problems on Windowing Method :Ex. 7.5.1 : Design a linear phase FIR low pass filter of length seven with cut-off frequency 1 rad/sec using rectangular window.Soln.:Step I: The desired frequency response Hd (co) for the low pass FIR filter is given by,

Step III: Here we have to make use of rectangular window of the order 7. We have for rectangular

Equation (8) gives unit impulse response of FIR filter. Making use of Equation (7), we can obtair the values of hd (n) and h (n) as shown in the Table P. 7.5.1.Table P. 7.5.1

...en

step III: Here we have to make use of rectangular window of the order 7. We have for rectangula window, (forn = 0toM-l

- nf FTR filter Making use of Equation (7), we can obta Equation (8) gives unit impulse "^Tt h^Ti

the values of hd(n) andh(n) as shown m the TableP. 7.5.2.

Making use of these equations we can obtain the values of h ( n) as shown in Table P. 7.5.3.Table P. 7.5.3

Magnitude and phase:

7.6 Design of IIR Filters using Approximation of Derivatives :Consider an analog differentiator with transfer function Ha (s). The function of analot differentiator is to take the derivative of analog input signal. Let x (t) be the input signal applied to analog differentiator. Then its output can be written as,

...(7.6.1

Since the output is the differentiation of input, then from Fig. 7.6.1(a) we can write,Equation (7.6.3) gives the transfer function of digital filter. Now we will obtain the transfer function of analog filter.Equation (7.6.4) gives the transfer function of analog filter.

...(7.6.2)...(7.6.3) ...(7.6.4)

Putting a = 0 inEquation (7.6.8) we get,

...(7.6.9)Now as Q varies from - to + , the corresponding locus of points in Z plane is a circle of radius ^ and its centre at Z = t. This mapping is shown in Fig. 7.6.1(b).

The mapping shows that,(i) L.H.S. of s plane is mapped onto the points inside the circle in Z plane having radius = ~z andcentre at Z = r. K(ii) R.H.S. of a plane is mapped onto the points outside the circle in Z plane,(hi) The stable analog filter is converted into stable digital filter.Limitation of approximation of derivatives method :This method is suitable only for designing of low pass and bandpass IIR digital filters wrelatively small resonant frequency.7.7 Impulse Invariant Method :In this method, the design starts from the specifications of analog filter. Here we have to replantanalog filter by digital filter. This is achieved if impulse response of digital filter resembles the samplerversion of impulse response of analog filter. If impulse response of both, analog and digital filtermatches then, both filters perform in a similar manner.Before studying this method we will list out the different notations, we are going to use. h (t) = Impulse response in time domain Ha ( s ) = Transfer function of analog filter; here V is Laplace operator

h (n Ts)=Sampled version of h (t), obtained by replacing t by n Ts.H (Z)=Z transform of h ( nTs). This is response of digital filter.Q=Analog frequencyco=Digital frequencyTransformation of analog system function Ha (s) to digital system function H (Z):Now let the system transfer function of analog filter be Ha ( s ). We can express Ha (s) in termsrvf nnrtial fm^tinri PYnaiTiinn Th^f mpans

...(7.7.1)

Here Ak - Ax, A2 ... AN are the coefficients of partial fraction expansion.and Pk = Pj, P2 ... PN are the poles.

Here V is the laplace operator. So we can obtain impulse response of analog filter, h (t) from Ha( s) by taking inverse laplace of Ha ( s ). So using standard relation of inverse laplace we get,...(7.7.2)

Now unit impulse response for discrete structure is obtained by sampling h ( t ). That means, h (n) can be obtained from h (t) by replacing 't' by n Ts in Equation (7.7.2).

...(7.7.3)Here Ts is the sampling time.,The system transfer function of digital filter is denoted by H ( Z ). It is obtained by taking Z - transform of h (n ). According to the definition of Z - transform for causal system,

...(7.7.4)

...(7.7.5)...(7.7.6)

This is the required transfer function of digital filter.Thus comparing Equations (7.7.1) and (7.7.6), we can say that the transfer function of digital filter is obtained from the transfer function of analoa filter bv doing the transformation.

Equation (7.5.7) shows, how the poles from analog domain are transferred into the dig domain. This transformation of poles is called as mapping of poles.7.7.1 Relationship of S-plane to Z-plane (Mapping between S-plane and Z-plane):We know that the poles of analog filters are located at s = Pk. Now from Equation (7.7.7) we cm say that the poles of digital filter, H (Z) are located at,Z = e k s...(7.7.8*,This equation indicates that the poles of analog filter at s = Pk are transformed into the pole? IPkTdigital filter at Z = e. Thus the relationship between laplace ( V domain) and Z domain is giv-sTsHere s = Pk and Ts is the sampling time. Now 's' is the laplace operator and it is expressed as,s = a + jQ...(7.7.HHere a = Attenuation factor and Q. = Analog frequency We know the 'Z' can be expressed in polar form as,Here 'r' is magnitude and 'co' is the digital frequency.

Putting Equations (7.7.10) and (7.7.11) in Equation (7.7.9) we get,

Separating real and imaginary parts of Equation (7.7.12) we get,

...(7.7.12)...(7.7.13) ...(7.7.14)

Now we will find the relationship between s plane and Z plane. Basically plot in 's'-domain means, a is plotted on X-axis and jQ) is plotted on Y- axis. And Z-domain representation means real Z is plotted on X- axis and imaginary Z is plotted on Y-axis.Now consider Equation (7.7.13), it is

(i) If o < 0, then r is equal to reciprocal of 'e' raise to some constant. Thus range of r will be 0 to 1.

p.. '.;.. ........... ............. ......... .........,.. . ............... ... .,.,...,;....(7.7.15)Now o < 0 means negative values of a. That is L.H.S. of s plane. We know that 'r' is the radius of circle is Z plane.So '0 < r < 1' indicates interior part of unit circle. Thus we can conclude that,L.H.S. of V plane is mapped inside the unit circle.(in Tf n = 0 then r = e = 1

Now a = 0 indicates jQ, axis and r = 1 indicates unit circle. Thus, jQ axis in 's' plane is mapped on the unit circle.(iii) If a > 0 then, r is equal to 'e' raise to some constant. That means r > 1.

Now a > 0 indicates R.H.S. of 's' plane and 'r' > 1 indicates exterior part of unit circle. Thus, R.H.S. of's' plane is mapped outside the unit circle.Combining all conditions; this mapping is shown in Fig. 7.7.1.

Fig. 7.7.1: Relationship of s plane to Z-plane Disadvantages of impulse invariance method :

this range also; '' maps from - n to n. Thus mapping from analog frequency 'Q,' to digffl frequency 'co' is many to one. This mapping is not one to one. (2) Analog filters are not band limited so there will be aliasing due to the sampling process. Because of this aliasing, the frequency response of resulting digital filter will not be identical tc t original frequency response of analog filter.7.7.2 Solved Problems using Impulse Invariance Method :Ex. 7.7.1 : Find out H ( Z ) using impulse invariance method at 5 Hz sampling frequency from H a as given below:

To convert each term into positive powers of Z ; multiplying numerator and denominator of each term by Z we get,

Ex. 7.7.2 : Consider a causal continuous time system with impulse response hc(f) and the syste-functionu / \s + aHc(s) = 7?zrz-cv ' (s + a) + bUse impulse invariance to determine H (z) for a discrete time system such that h(n) = hc(nT).

I

7.8 Bi-linear Transform Filter (BLT):In case of impulse invariance method, we have studied that the mapping is many to one. So this method is not suitable to design high-pass filter and band reject filter.In case of bilinear transformation; the mapping is one to one from s domain to the Z domain. So there is no aliasing effect. The limitations of impulse invariance method are overcome by using BLT method.Relationship of s plane and Z plane :Thus relationship between s plane and Z-plane is given by,

Now we will discuss the following conditions related to Equation (7.8.7).(i) When r < 1 then a < 0Here r < 1, means interior part of circle having unit circle and a < 0, means o is negative which is L.H.S. of s-plane. So this condition indicates that L.H.S. of s plane maps inside the unit circle.(ii) When r = 1 then a = 0Now r = 1 means unit circle and a = 0 means jQ axis. Thus this condition indicates that the jQ axis maps on the unit circle.(iii) When r > 1 then a > 0Here r > 1, means exterior part of unit circle and a > 0 indicates that a is positive means R.H.S. of s-plane. So this condition indicates that R.H.S. of s-plane maps outside the unit circle.This mapping is similar to the mapping in impulse invariance method.But in impulse invariance method mapping is valid only for poles ; while in bilinear transformation, mapping is valid for poles as well as zeros.How Stable Analog Filter is Converted into Stable Digital Filter ?Analog filter is stable if the poles lie on the L.H.S. of s-plane. While the digital filter is stable if the poles are inside the unit circle in the Z-domain. Now condition (i) indicates that L.H.S. of s-plane maps inside the unit circle. Thus stable analog filter is converted into stable digital filter.Advantages of Bilinear Transformation Method :1. There is one to one transformation from the s-domain to the Z- domain.2. The mapping is one to one.3. There is no aliasing effect.4. Stable analog filter is transformed into the stable digital filter.5. 7.8.1 Disadvantage of Bilinear Transformation Method :The mapping is non-linear and because of this; frequency warping effect takes place.7.9 Comparison between Impulse Invariance and Bilinear TransformationMethod :

7.10 Solved Problems using BLT :...(1) (2) (3)Ex. 7.10.1: An analog filter has the following transfer function H (s) = ^rpf- Using bilineartransformation technique, determine the transfer function of digital filter H (Z) and also write the difference equation of digital filter.Soln.: The given transfer function is,

Ex. 7.10.3 : Design a single pole low pass digital filter with a 3 dB bandwidth of 0.2 n by use of bilinear transformation applied to the analog filter.

Soln.: The given transfer function of analog filter is,

-(I)

Thus we have obtained the transfer function of analog filter in the appropriate form. Now to obtain H (Z) we have to put,

...(5)This is the required transfer function for digital filter. Note that there is a single pole at P, = 0.509.Now in the problem frequency of digital filter is given which is coc = 0.2 n and bandwidth is 3 dB. We will check the frequency response. The frequency response of digital filter is obtained by putting Z = eJ(I> in the equation of H (Z). Thus we get,

...(6)Ex. 7.10.4 : The system transfer function of analog filter is given by,

Obtain the system transfer function of digital filter using BLT which is resonant at cor = 5

Ex. 7.10.6: The analog transfer function of low pass filter is, H ( s ) = ^T^ and its temdwidt- = 1 rad/sec.Design the digital filter using BLT method whose cut-off frequency is 20 jt and samp -; time is 0.0167 sec; by considering the warping effect.

Ex. 7.10.7 : Design a digital low pass MR filter to approximate the following transfer function.-4

Using the bilinear transformation method, obtain the transfer function H(Z) of digital filter assuming 3 dB cut-off frequency of 150 Hz and sampling frequency of 1.28 KHz.Soln.: Given,Cut-off freauencv F =150 Hz, Sampling frequency F = 1.28 KHz

7.11 Basic Filter Approximations :We have studied that the digital IIR filters are designed from the analog filters. Many times it is necessary to approximate the characteristics of analog filter. This approximation is required because the practical characteristic of a filter is not identical to the ideal characteristics. There are three different types of approximation techniques as follows :1. Butterworth filter approximation.2. Chebyshev filter approximation.3. Elliptic filter approximation.7.11.1 Butterworth Filter Approximation :A typical characteristic of a butterworth low pass filter is as shown in Fig. 7.11.1. This type of response is called as butterworth response because its main characteristic is that the passband is maximally flat. That means there are no variations (ripples) in the passband. Now the magnitude squared response of low pass butterworth filter is given bv.

...(7.11.1

Fig. 7.11.1: Typical characteristics of analog L.P.F.This equation is also expressed as,Here | H (Q.) |=Magnitude of analog low pass filter.Oc=Cut-off frequency (- 3B frequency).Q.-Pass band edge frequency.1 + e=Pass band edge value.1+8=Stop band edge value.6=Parameter related to ripples in pass band.8=Parameter related to ripples in stop band.N=Order of the filter.We know that, in case of low pass filter the frequencies will pass upto the value of cut-off frequency (Qc). This is called as pass band. After that the frequencies are attenuated. This is called as stop band. Ideal characteristic is shown by dotted line in Fig. 7.11.2. Ideally, at the value of cut-off frequency (lc) the frequencies should be stopped. But in practical cases this is not happening.Now the order of filter is denoted by 'N'. Roughly we can say order of filter means, the number of stages used in the design of analog filter. As the order of filter 'N' increases, the response of filter is more close to the ideal response as shown in Fig. 7.11.2.

Fig. 7.11.2 : Effect of N on frequency response characteristicsSalient features of low pass butterworth filter:1. The magnitude response is nearly constant (equal to 1) at lower frequencies. That means pa band is maximally flat.2. There are no ripples in the pass band and stop band.3. The maximum gain occurs at Q. = 0 and it is | H (0) | = 1.4. The magnitude response is monotonically decreasing.7.11.2 Chebyshev Filter:Using chebyshev filter design, there are two subgroups,1. Type-1 chebyshev filter2. Type-2 chebyshev filter1. Type-1 Chebyshev Filter:These filters are all pole filters. In the passband, these filters show equiripple behaviour and the> have monotonic characteristics in the stopband.The filter characteristics for odd and even values of N are shown in Fig. 7.11.3. The magnitude squared frequency response is given by,

Here e = Ripple parameter in the passband; Qp = Passband frequency CN (x) = Chebyshev polynomial of order N....(7.11.3

Fig. 7.11.3 : Type-1 chebyshev filter characteristicsThe chebyshev polynomials are determined by using the equation,CN + 1(x) = 2xCN(x)-CN_1(x)...(7.11.4)With C0 (x) = 1 and Q (x) = x The different polynomials are given in Table 7.11.1.Table 7.11.1

The major difference between butterworth and chebyshev filter is that the poles of butterworth filter lie on the circle, while the poles of chebyshev filter lie on ellipse.2. Type-ll Chebyshev Filter [Inverse Chebyshev Filters]:This filter contains zeros as well as poles. The characteristic of such filters for even and odd values of N is shown in Fig. 7.11.4. The magnitude squared response is given by,

HereCN (x) = N* order polynomial;Qs = Stopband frequency; Qp = Passband frequency

7.11.3 Elliptic Filters:Elliptic filters are also called as cauer filters. Such filters have equiripple passband and stopband The magnitude squared frequency response of elliptic filter is shown in Fig. 7.11.5.The maenitude sauared resoonse of elliDtic filter is obtained bv usine.

Here = Ripple parameter in passband Qp = Passband edge frequency TJN (x) = Jocobian elliptic function of order N.

The order 'N" of elliptic filter is given by,Here 8 is related to 5, and 52 as follows :52=a/1 + S282=stopband rippleand S,=101ogi0(l +e )where 8j=stopband rippleThe function K (x) is called as elliptic integral of first kind. Advantages of elliptic filters :1. Elliptic filter is more efficient because it provides the smallest order filter for given set of specifications.2. It has smallest transition bandwidth.Disadvantages of elliptic filters :The phase response is more non-linear in passband.7.12 Frequency Transformations :Uptill now we have studied the design of low pass filter only. Now if it is asked to design other filter like high pass, band pass or band reject filter then we have to use the frequency transformation.If the cut-off frequency of LPF is equal to 1 that means if Qc = 1 then, it is called as normalizedfilter. To design the other types of filters; first the system function of normalized LPF is obtained. Then using frequency transformation we can get the system function of the required filter. The following formulae are used for the frequency transformation.Let, we have a normalized L.P.F. having cut-off frequency Qc.1. Low pass to low pass :Suppose it is asked to design another LPF with new passband edge frequency 2LP. Then use the transformation,

2. Low pass to high pass :Suppose we have to design HPF with cutoff frequency 2HP then use,i'. 'iI

3. Low pass to bandpass :Suppose we have to design bandpass filter with higher cutoff frequency Qu and lower cut-off frequency 2[ then use the transformation.

4. Lowpass to bandstop or notch or band reject:Suppose we have to design notch filter with higher cut-off frequency 2U and lower cutoff frequency 2, then use the transformation,

Note : These formulae are applicable for analog frequency transformation. Similarly, we cantransform