2.161 Signal Processing: Continuous and Discrete Fall 2008

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1 MI OpenCourseWare Signal Processing: Continuous and Discrete Fall 008 For information about citing these materials or our erms of Use, visit:

2 Massachusetts Institute of echnology Department of Mechanical Engineering.6 Signal Processing - Continuous and Discrete Fall erm 008 Lecture 8 Reading: Proakis and Manolakis: 7.3., 7.3., 0.3 Oppenheim, Schafer, and Buck: 8.7.3, 7. FF Convolution for FIR Filters he response of an FIR filter with impulse response {h k } to an input {f k } is given by the linear convolution y n = f k h n k. k= he length of the convolution of two finite sequences of lengths P and Q is N = P + Q. he following figure shows a sequence {f n } of length P = 6, and a sequence {h n } of length Q = 4 reversed and shifted so as to compute the extremes of the convolved sequence y 0 and y 8. B $ # D 3 " &! D 3 " & he convolution property of the DF suggests that the FF might be used to convolve two equal length sequences y n = IDF {DF {f n }.DF {h n }}. copyright c D.Rowell 008 8

3 However, DF convolution is a circular convolution, involving periodic extensions of the two sequences. he following figure shows the circular convolution of length 6, on two sequences {f n } of length P = 6 and {h n } of length Q = 4. he periodic extensions cause overlap in the first Q samples, generating wrap-around errors in the DF convolution. B # D $! 3 " L A H = F B 3 I = F A I DF convolution of two sequences of length P and Q (P Q) in DFs of length P. Produces an output sequence of length P, whereas linear convolution produces an output sequence of length P + Q.. Introduces wrap-around error in the first Q samples of the output sequence. he solution is to zero-pad both input sequences to a length N P +Q and then to use DF convolution with the length N sequences. For example, if {f n } is of length P = 37, and {h n } is of length Q = 5, for error-free convolution we must perform the DFs in length N = 46. If the available FF routine is radix-, we should choose N=5. he use of the FF for Filtering Long Data Sequences: he DF convolution method provides an attractive alternative to direct convolution when the length of the data record is very large. he general method is to break the data into manageable sections, then use the FF to to perform the convolution and then recombine the output sections. Care must be taken, however, to avoid wrap-around errors. here are two basic methods used for convolving long data records. Let the impulse response {h n } have length Q. Overlap-Save Method: (Also known as the overlap-discard, or select-savings method.) In this method the data is divided into blocks of length P samples, but with successive blocks overlapping by Q. he DF convolution is done on each block with length P, and wrap-around errors are allowed to contaminate the first Q samples of the output. hese initial samples are then discarded, and only the error-free P (Q ) samples are saved in the output record. 8

4 With the overlap of the data blocks, in the mth block the samples are f m (n) = f(n + m(p (Q ))), n = 0,...,P, and after DF convolution in length P, giving y mp (n), the output is taken as { y mp (n +(Q )), n =0,...,P (Q ) y m (n) = 0, otherwise. and the output is formed by concatenating all such records: y(n) = y m (n m(p Q + )). m=0 D 3 B L A H = F F E C E F K J I A? J E I E I? = K J F K J I = F A I E J D E I H A C E 3 E I? = K J F K J I = F A I E J D E I H A C E 3 E I? = K J F K J I = F A I E J D E I H A C E 3 Overlap-Add Method: In this method the data is divided into blocks of length P, but the DF convolution is done in zero-padded blocks of length N = P + Q so that 8 3

5 wrap-around errors do not occur. In this case the output is identical to the linear convolution of the two blocks, with an initial rise of length Q samples, and a trailing section also of length Q samples. It is easy to show that if the trailing section of the mth output block is overlapped with the initial section of the (m + )th block, the samples add together to generate the correct output values. D 3 B K J F K J I = F A I E J D E I H A C E 3 K J F K J I = F A I E J D E I H A C E 3 O! 3 MALAB s fftfilt() function performs DF convolution using the overlap-add method. he Design of IIR Filters An IIR filter is characterized by a recursive difference equation and a rational transfer function of the form N M y n = a k y n k + b k f n k k= k=0 b 0 z 0 + b z b M z M H(z) = z0 + a z a N z N IIR filters have the advantage that they can give a better cut-off characteristic than a FIR filter of the same order, but have the disadvantage that the phase response cannot be well controlled. 8 4

6 he most common design procedure for digital IIR filters is to design a continuous filter in the s-plane, and then to transform that filter to the z-plane. Because the mapping between the continuous and discrete domains cannot be done exactly, the various design methods are at best approximations.. Design By Approximation of Derivatives: Perhaps the simplest method for low-order systems is to use backward-difference approximation to continuous domain derivatives. Example Suppose we wish to make a discrete-time filter based on a prototype first-order high-pass filter s H p (s) =. s + a he differential equation describing this filter is dy df + ay = dt dt he backward-difference approximation to a derivative based on samples taken at intervals apart is dx x n x n dt and substitution into the differential equation gives y n y n f n f n + ay n = or y n = y n + (f n f n ) + a + a he transfer function is z z H(z) = = ( + a ) + z ( + a )z + his example indicates that the method uses the transformation in H p (s). For higher order terms z s ( ) z n n s. 8 5

7 Example Convert the continuous low-pass Butterworth filter with Ω c = rad/s to a digital filter with a sampling time =0.5 s. he transfer function is he discrete-time transfer function is and with =0.5 s, H p (s) =. s + s + H(z) = ( ) ( ) z z + + = ( + + ) ( + )z + z 0.5 H(z) = z + z he frequency response of this filter is plotted in Example 4. In general, the backward-difference does not lead to satisfactory digital filters that mimic the prototype filter characteristics. (See Proakis and Manolakis, Sec. 0.3.).. Design by Impulse-Invariance: In the impulse-invariant design method the impulse response {h n } of the digital filter is taken to be proportional to the samples of the impulse response h p (t) of the continuous filter H p (s) with a sampling interval of seconds. he most common form is h n = h p (n ). D J J H J JO F A 0 I D J I = F A H D 6 J 6 8 6

8 hen { H(z) = Z {h p (n )} = Z L {H p (s)} } since h p (t) = L {H p (s)}, and where Z {} indicates the z-transform of a continuous function with sampling interval. Example 3 Find the impulse-invariant IIR filter from the prototype continuous filter Solution: a H p (s) =. s + a Using Laplace transform tables { } h p (t) = L a s + a = a e at. and from z-transform tables { } a a e at Z =. e a z he IIR filter is and the difference equation is { } a e at H(z) = Z = y n = e a y n + a f n a e a z For the digital filter H( e j Ω ) = H(z) j Ω = h k e z= e j kω k=0 and the DF of the samples of the continuous prototype s impulse response is ( ( πk )) DF {h p (n )} = h p (k ) e j kω = Hp j Ω. k=0 k= hen if h n = h p (n ), ( ( πk )) H( e j Ω ) = H p j Ω. k= he discrete-time frequency response is therefore a superposition of shifted replicas of the frequency response of the prototype. As a result, aliasing will be present in H( e j ω ) if the prototype s frequency response H p (j Ω) = 0 for Ω π/. 8 7

9 F F 0 A M L A H = F F E C H A F E? = I = E = I E E I J H J E E J D A F = I I > F F M For this reason the impulse-invariance method is not suitable for the design of high-pass or band-stop filters, which by definition require a prototype H p (s) with a non-zero frequency response at Ω = π/. Example 4 Design an impulse-invariant filter based on the second-order low-pass Butterworth prototype used in Example, with = 0.5 s. H p (s) = s + s + Solution: From z-transform tables { { }} β e a sin(β )z Z L = (s + a) + β z + z e a cos(β )z + e a and H p (s) may be written in this form H p (s) = (s + / ) + (/ ) so that a = /, and β = /. Substituting these values, 0.79z H(z) = Z { L {H p (s)} } =.375z z he frequency response of the impulse-invariant, and backward-difference (from Example ) filters are compared with the prototype below: 8 8

10 ' N H J J O F A * K J J A H M H J D. E J A H. H A G K A? O H A I F I A = C E J A & % $ # "! F K I A E L = H E = J 6 # * =? M E B B A H A? A 6 #! " # $. H A G K A? O H I he MALAB function [bz, az] = impinvar(bs, as, Fs) will compute the numerator az, and denominator bz coefficients for an impulse-invariant filter from the continuous prototype coefficients bs and as, with a sampling frequency Fs. he filter in Example 4 can be designed in a single line: [bz, az] = impinvar(, [ sqrt() ], ). 8 9