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Digital Signal Processing

Digital Signal Processing

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Digital Signal Processing

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  1. Digital Signal Processing Prof. V. N. Bhonge Dept. of Electronics & Telecomm. Shri Sant Gajanan Maharaj College of Engg, Shegaon – 444203

  2. 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

  3. 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.

  4. 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.

  5. 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:

  6. 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

  7. 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.

  8. 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).

  9. 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)

  10. (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.

  11. 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.

  12. 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

  13. 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

  14. 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.

  15. (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

  16. 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.

  17. 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:

  18. (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.

  19. 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 = 2150 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

  20. 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.

  21. Solution: The cut-off frequency of the digital filter is wp = 230 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:

  22. 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:

  23. Using the lowpass-to-bandpass transformation, equation (16), we have Applying the BLT gives

  24. 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:

  25. Main Property of the Bilinear Transformation: it preserves the stability regions. s-plane z-plane since:

  26. Mapping of Frequency with the Bilinear Transformation. Magnitude: Phase: where

  27. See the meaning of this: it is a frequency mapping between analog frequency and digital freqiency.

  28. 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.

  29. Design of Analog Filters Specifications: pass band transition band stop band

  30. SSGMCE Shegaon Dept. of E & T Prof. V. N. Bhonge

  31. SSGMCE Shegaon Dept. of E & T Prof. V. N. Bhonge

  32. SSGMCE Shegaon Dept. of E & T Prof. V. N. Bhonge

  33. SSGMCE Shegaon Dept. of E & T Prof. V. N. Bhonge

  34. 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

  35. 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

  36. 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

  37. Properties of an IIR Filter SSGMCE Shegaon IIR Equation IIR structure for N = M = 2 Dept. of E & T Prof. V. N. Bhonge

  38. 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

  39. Filter Specification - Step 1 SSGMCE Shegaon Dept. of E & T Prof. V. N. Bhonge

  40. 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

  41. 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

  42. 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

  43. 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

  44. Two Major Techniques: Butterworth, Chebychev Butterworth: Specify from passband, determine N from stopband:

  45. Poles of Butterworth Filter: which yields the poles as solutions and choose the N poles in the stable region. N=2 + + poles + + s-plane

  46. 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

  47. 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

  48. 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.

  49. Chebychev Filter: Since (easy to show from the definition), then

  50. 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