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## Digital Signal Processing

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**Digital Signal Processing**Prof. V. N. Bhonge Dept. of Electronics & Telecomm. Shri Sant Gajanan Maharaj College of Engg, Shegaon – 444203 vnbhonge@rediffmail.com**UNIT-II**Infinite Impulse Response (IIR) filters: IIR filter design by approximation of derivatives, impulse invariant method, Bilinear transformation method, Butterworth filter and Chebyshev filter. Realization of basic structure IIR system: Direct form I, Direct form II, Cascade and parallel. SSGMCE Shegaon Dept. of E & T Prof. V. N. Bhonge**Infinite Impulse Response (IIR) Filters**Recursive Filters: with constant coefficients. Advantages:very selective filters with a few parameters; Disadvantages: a) in general nonlinear phase, b) can be unstable.**analog**digital s-plane z-plane Design Techniques:discretization of analog filters Problem: we need to map the derivative operator “s” into the time shift operator “z”, and make sure that the resulting system is still stable.**Two major techniques**• Euler Approximation (easiest), • Bilinear Transformation (best). Euler Approximation of the differential operator: approximation of “s” take the z-Transform of both sides:**analog**digital s-plane z-plane Example: take the analog filter with transfer function and discretize it with a sampling frequency . By Euler’s approximation The filter is implemented by the difference equation**Problem with Euler Approximation:**it maps the whole stable region of the s-plane into a subset of the stable region in the z-plane s-plane z-plane since if Re[s]<0.**IIR Filter Design by Approximation of Derivatives**Analogue filters having rational transfer function H(s) can be described by the linear constant coefficient differential equation. One of the simplest methods for converting an analog filter into a digital filter is to approximate the differential equation by an equivalent difference equation. This approach is often used to solve a linear constant coefficient differential equation numerically on a digital computer. For the derivative dy(t)/dt at a time t = nT we substitute the backward difference [y[n]-y[n-1)]/T. Thus (1) Where T represents the sampling interval and y(n) = y(nT).**The analogue differentiator with output dy(t)/dt has the**system function H(s) = s, while the digital system that produces the output [y(n)-y(n-1)]/T has the transfer function H(s) = (1 – z-1)/T. Consequently, the frequency domain equivalent is: (2) The second derivative d2y(t)/dt2 is replaced by the second difference which is given by (3) In the frequency domain, (3) is equivalent to (4)**(5)**Implications: From (2) we have It is easy to generalize equation (2) and (4) as (6) If we substitute s = jw in (6), we have (7) As w varies from - to + , the corresponding locus of points in the z-plane is a circle of radius ½ and with centre at z =1/2, as shown in the figure given on the next slide.**jw**s-plane 1/2 The above figure shows that the mapping in (7) takes points in the LHP of s into corresponding points inside this circle (i.e. circle with radius ½ and the points in RHP plane in s are mapped into points outside this circle. Consequently, this mapping has the desirable property that a stable analog filter is Transformed into a stable digital filter. However, the possible location of the poles of the digital filter are confined to relatively small frequencies and, thus, the mapping is restricted to low-pass and band-pass filters having relatively small resonant frequencies.**Example1:Convert the analogue band-pass filter with system**functiongiven below into a digital IIR filter by use of the backward difference for the derivative. Solution: substituting s = (1-z-1)/T yields**Bilinear Transformation (Tustin’s Method)**This is the most widely used transformation which is suitable for the design of low-pass, high-pass, band-pass and band-stop filters. In the Bilinear Transformation (BLT) method, the basic operation required to convert an analogue filter is to replace s as follows: (8) k = 1 or 2/T**To investigate the characteristics of the bilinear**transformation, let z = rejwT s = + jw’ Now Consequently, (9) (10) Note that if r < 1, <0, and if r > 1, > 0. Consequently, the LHP in s maps into the inside of the unit circle in the z-plane and the RHP in s maps into the outside of the unit circle.**(11)**or (12) The relationship between the frequency variables in the two domains is illustrated in the following figure: When r = 1, then = 0 and 20 18 16 14 12 w’ 10 8 6 4 2 w 0 0 0.5 1 1.5 2 2.5 3 3.5**From the figure we observe that the mapping between w and**w’ is almost linear for small values of w, but becomes non-linear for larger values of w, leading to a distortion (or warping) of the digital frequency response. This effect is normally compensated for by pre-warping the the analog filter before applying the bilinear transformation. To compensate for the effect, we pre-warp one or more critical frequencies before applying the BLT. For example, for a low-pass filter, we often pre-warp the cutoff or bandedge frequency as follows: (13) where wp = specified cutoff frequency w’p = prewarped cutoff frequency k = 1 or 2/T T = sampling period.**Summary of the BLT method of coefficient calculation:**For standard, frequency selective IIR filters, the steps for using the BLT method may be summarized as follows: • Use the digital filter specifications to find a suitable normalized, prototype, analogue lowpass filter, H(s). • Determine and prewarp the bandedge or critical frequencies of the desired filter. • Denormalize the analog prototype filter by replacing s in the transfer function H(s), using one of the following transformations, depending on the type of filter required:**(14)**lowpass to lowpass lowpass to highpass (15) lowpass to bandpass (16) lowpass to bandstop (17) where and • Apply the BLT to obtain the desired digital filter transfer • function.**Example 1:Design a digital low-pass filter to approximate**the following transfer function: Using the BLT method obtain the transfer function, H(z), of the digital filter, assuming a 3 dB cutoff frequency of 150 Hz and a sampling frequency of 1.28kHz. Solution: wp = 2150 rad/sec, T = 1/Fs = 1/1280, giving a prewarped critical frequency of w’p = tan(wpT/2) = 0.3857. The frequency scaled analog filter is given by**Applying the BLT gives**Example 2: The normalized transfer function of a simple, analog lowpass, filter given by Starting from the s-plane, determine, using the BLT method, the transfer function of an equivalent discrete time high-pass filter. Assume a sampling frequency of 150 Hz and a cutoff frequency of 30 Hz.**Solution: The cut-off frequency of the digital filter is wp**= 230 rad/sec. The cut-off frequency, after prewarping is w’p = tan(wpT/2). With T = 1/150, w’p = tan(/5) = 0.7265. Using the low to high-pass transformation of equation (15), the denormalized analog transfer function is obtained as The z-transfer function is obtained by applying the BLT:**Example 3: A discrete time bandpass filter with Butterworth**characteristics meeting the specifications given below is required. Obtain the coefficients of the filter using the BLT method. passband 200 – 300 Hz sampling frequency 2 kHz filter order 2 Solution: A first-order normalized analog low-pass filter is required (since the frequency band transformation for for bandpass filters – equation (16) – will double the filter order. Thus The prewarped critical frequencies are:**Using the lowpass-to-bandpass transformation, equation**(16), we have Applying the BLT gives**C**B A D nT-T nT Bilinear Transformation. It is based on the relationship Take the z-Transform of both sides: which yields the bilinear transformation:**Main Property of the Bilinear Transformation: it preserves**the stability regions. s-plane z-plane since:**Mapping of Frequency with the Bilinear Transformation.**Magnitude: Phase: where**See the meaning of this:**it is a frequency mapping between analog frequency and digital freqiency.**Example:we want to design a digital low pass filter with a**bandwith and a sampling frequency . Use the Bilinear Transformation. Solution: • Step 1: specs in the digital freq. domain • Step 2: specs of the analog filter to be digitized: or equivalently • Step 3: design an analog low pass filter (more later) with a bandwith ; • Step 4: apply Bilinear Transformation to obtain desired digital filter.**Design of Analog Filters**Specifications: pass band transition band stop band**SSGMCE**Shegaon Dept. of E & T Prof. V. N. Bhonge**SSGMCE**Shegaon Dept. of E & T Prof. V. N. Bhonge**SSGMCE**Shegaon Dept. of E & T Prof. V. N. Bhonge**SSGMCE**Shegaon Dept. of E & T Prof. V. N. Bhonge**Introduction**SSGMCE Shegaon • Infinite Impulse Response (IIR) filters are the first choice when: • Speed is paramount. • Phase non-linearity is acceptable. • IIR filters are computationally more efficient than FIR filters as they require fewer coefficients due to the fact that they use feedback or poles. • However feedback can result in the filter becoming unstable if the coefficients deviate from their true values. Dept. of E & T Prof. V. N. Bhonge**Properties of an IIR Filter**SSGMCE Shegaon • The general equation of an IIR filter can be expressed as follows: • ak and bk are the filter coefficients. Dept. of E & T Prof. V. N. Bhonge**The transfer function can be factorised to give:**SSGMCE Shegaon • Where: z1, z2, …, zN are the zeros, p1, p2, …, pN are the poles. • For the implementation of the above equation we need the difference equation: Dept. of E & T Prof. V. N. Bhonge**Properties of an IIR Filter**SSGMCE Shegaon IIR Equation IIR structure for N = M = 2 Dept. of E & T Prof. V. N. Bhonge**Design Procedure**SSGMCE Shegaon • To fully design and implement a filter five steps are required: (1) Filter specification. (2) Coefficient calculation. (3) Structure selection. (4) Simulation (optional). (5) Implementation. Dept. of E & T Prof. V. N. Bhonge**Filter Specification - Step 1**SSGMCE Shegaon Dept. of E & T Prof. V. N. Bhonge**Coefficient Calculation - Step 2**SSGMCE Shegaon • There are two different methods available for calculating the coefficients: • Direct placement of poles and zeros. • Using analogue filter design. • Both of these methods are described. Dept. of E & T Prof. V. N. Bhonge**Placement Method**SSGMCE Shegaon • All that is required for this method is the knowledge that: • Placing a zero near or on the unit circle in the z-plane will minimise the transfer function at this point. • Placing a pole near or on the unit circle in the z-plane will maximise the transfer function at this point. • To obtain real coefficients the poles and zeros must either be real or occur in complex conjugate pairs. Dept. of E & T Prof. V. N. Bhonge**Analogue to Digital Filter Conversion**SSGMCE Shegaon • This is one of the simplest method. • There is a rich collection of prototype analogue filters with well-established analysis methods. • The method involves designing an analogue filter and then transforming it to a digital filter. • The two principle methods are: • Bilinear transform method (\Links\Bilinear Theory.pdf). • Impulse invariant method. Dept. of E & T Prof. V. N. Bhonge**Bilinear Transform Method**SSGMCE Shegaon • Practical example of the bilinear transform method: • The design of a digital filter to approximate a second order low-pass analogue filter is required. • The transfer function that describes the analogue filter is: • The digital filter is required to have: • Cut-off frequency of 6kHz. • Sampling frequency of 20kHz. Dept. of E & T Prof. V. N. Bhonge**Two Major Techniques: Butterworth, Chebychev**Butterworth: Specify from passband, determine N from stopband:**Poles of Butterworth Filter:**which yields the poles as solutions and choose the N poles in the stable region. N=2 + + poles + + s-plane**Example: design a low pass filter, Butterworth, with 3dB**bandwith of 500Hz and 40dB attenuation at 1000Hz. Solution: solve for N from the expression poles at**Chebychev Filters.**Based on Chebychev Polynomials: Property of Chenychev Polynomials: within the interval Chebychev polynomials have least maximum deviation from 0 compared to polynomials of the same degree and same highest order coefficient**root**root root Why? Suppose there exists with smaller deviation then But: has degree 2 … … and it cannot have three roots!!! So: you cannot find a P(x) which does better (in terms of deviation from 0) then the Chebychev polynomial.**Chebychev Filter:**Since (easy to show from the definition), then**Design of Chebychev Filters:**Formulas are tedious to derive. Just give the results: Given: the passband, and which determines the ripple in the passband, compute the poles from the formulae s-plane