Signal Processing and Linear Systems-B.P.Lathi copy

# V e xercise e io7 t he i nput i k o f a n l tio s

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Unformatted text preview: hown i n F ig. 10.17. We now convolve e ach o f t hese b locks w ith h[k] ( padded b y L - 1 zeros). A s b efore, D FT is u sed t o p erform c onvolution. - :he o utput s equence c orresponding t o e ach b lock also h as a l ength L + M - 1. W e d iscard t he f irst M - 1 d ata p oints a nd s ave t he l ast L d ata p oints f rom e ach o utput b lock, a s d epicted i n F ig. 10.17. T he t otal o utput is given b y c ombining a ll t he s aved b locks in sequence. T his m ethod is known a s t he overlap a nd save m ethod. M ore d etails o f t his m ethod c an b e f ound i n t he l iterature. I ••• F ig. 1 0.18 An Example of overlap and save method of block filtering. L D iscard 6 1 101101812121212 1 4 We follow t he procedure given in Fig. 10.17 t o section t he i nput d ata as shown in Fig. 10.18. Note t hat t he first M - 1 = 1 d ata p oint of t he first block is padded (zero) a nd t he l ast M _ 1 = 1 point of each block also appears as t he first point of t he n ext block. Each block of length L - M + 1 = 4 is now convolved (using D FT) w ith h[kJ ( padded with two zeros). The D FT procedure is already explained in Example 10.13. We shall omit t he d etails here. t he resulting o utput blocks are depicted in Fig. 10.18. T he first M - 1 = 1 p oint of each o utput block is discarded. T he t otal o utput is given by combining all t he saved blocks in sequence. • &lt;::) C omputer E xample C 10.6 Use MATLAB t o do example 10.13 (overlap a nd add m ethod). Here, we use t he MATLAB command 'fftfilt(h,f,M), t o p erform convolution using overlap and add method with blocks of length M . T his m-file is available in Signal P rocessing Toolbox. f =[3 2 3 1 0 1 0 1 l ]j h =[22]j L =length(f)+length(h)-lj k =0:1:L-2j y =fftfilt(h,f,3); s tem(k,rea1(y»j &lt;::) t::. E xercise E IO.8 The input I [k} of an LTID system is a sequence 1, 0, - 1, 2, . .. , and h[k}, the impulse response of the system is a 3-point sequence 3, 2, 3. Both sequences start at k = O. Using block convolution with L = 2, show t hat the output is 3, 2, 0, 4, . ... Derive your answer using both methods of block filtering. \l 1 0.7 Generalization o f t he D TFT t o t he Z- Transform L TID s ystems c an b e a nalyzed u sing D TFT. T his m ethod, h owever, h as t he • E xample 1 0.14 Using ove~lap and save method of block filtering, find t he response y[kJ of a n LTID system, whose Impulse response h[k] a nd t he i nput are illustrated in Fig. 10.16. following l imitations: 1. E xistence o f t he D TFT is g uaranteed o nly for a bsolutely s ummable s ignals [see 6~0 10 Fourier Analysis of Discrete-Time Signals 661 10.7 Generalization of t he D TFT t o t he z -transform z plane 1m Eq. (1O.36)J. T he D TFT does not exist for exponentially growing signals. This means t he D TFT m ethod can be applied only for a limited class o f i nputs. 2. Moreover, this method can b e applied only to asymptotically stable systems; it cannot b e used for unstable or even marginally stable systems. These are serious limita...
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