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Digital Signal ProceSSing

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ProblemS anD SolutionS Problem 15.1 In Figure P15.1-1 is shown the frequency response of a digital filter.(a) Determine and sketch the analog frequency response characteristics which, without aliasing will map to this digital frequency response when the impulse invariant transformation is applied.

Solution 15.1 In the absence of aliasing, the transformation from analog to digital frequency response corresponding to impulse invariance is Thus, in addition to a scale factor of l/T there is a linear mapping between analog and digital frequency given by

The desired analog frequency response can be obtained by reflecting the digital frequency response through this transformation, as indicated in Figure S15.1-1

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Figure S15.1-1 Note that since the frequency transformation between w and Q is linear, the shape of the frequency response is preserved. (b) Sketch the analog frequency response that will map to this digital frequency response when the bilinear transformation is applied.

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Figure P15.1-1 (b) For the bilinear transformation the transformation between analog and digital frequency is given by Thus, as in (a) we obtain the corresponding analog frequency response by reflecting the digital frequency response through this transformation as indicated in Figure S15.1-2.

Figure S15.l-2

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In this case, since the frequency transformation between w and Q is not linear, the linear slope of the digital frequency response does not correspond to a linear slope in the analog frequency response.

Problem 15.2 Repeat problem 15.1 for the digital frequency response characteristic in Figure P15.2-1.

Figure P15. 2-1 Solution 15.2 Figures S15.2-1 and S15.2-2 show the result of reflecting this frequency response characteristic through the frequency transformation for impulse invariance and the bilinear transformation respectively. In this case, since the digital frequency response is piecewise constant, its shape is preserved in the analog frequency response for both cases.

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Figure S15.2-1

Figure S15.2-2

Problem 15.3

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A digital lowpass filter is to be designed using the bilinear transformation. The specifications are that:

Determine the specifications on an analog low pass filter which will yield the desired digital filter when the bilinear transformation is applied with T = 1. 1 Solution 15.3 Since the filter is to be designed using the bilinear transformation with T = 1, the relation between analog and digital frequency is Thus we require that:

Problem 15.4

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An analog high pass filter can be obtained from an analog low pass filter by replacing s by 1/s in the transfer function: i.e., if Ga(s) is the transfer function for a lowpass filter, then Ha(s) is the transfer function for a high pass filter if

Also, a digital filter can be obtained by mapping an analog filter bymeans of the bilinear transformation . [Assume for convenience that T = 2 in Eq. (5.22) of the text.]

Figure P15.4-1 This mapping preserves the character of the magnitude characteristic, although the frequency scale is distorted. The network in Figure P 15.4-1 depicts a low pass filter with cut off frequency wL = 7/2.

The constants A,B,C, and D are real. Determine how to modify the coefficients in the network to obtain a high pass filter with cut off frequency WH = 7/2.

Solution 15.4

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Since Ga(s) is an analog low pass filter, G(z) given by

is a digital lowpass filter. Also, since Ha(s) = Ga(l/s) is an analog highpass filter,

Comparing H(z) with G(z) we observe that

Thus, multiplying the output of each delay by -l will convert a digitallowpass filter to a digital highpass filter. To determine the relationship between the cutoff frequencies, since e "=-1,

Thus the frequency response of the highpass filter is equal to thatof the lowpass filter, shifted by 7 as illustrated in Figure S15.4-1.

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Figure S15.4-1 The cutoff frequency of the high pass filter is (ff-w 2 ) and thus with is also as desired.

Finally, to determine how to modify each of the coefficients we notethat multiplying the output of each delay by -l is equivalent to changing the sign of all coefficient branches whose input has passed through an odd number of delays. Thus the coefficients A,C and the twccoe fficients of 2 are multiplied by -1.

Problem 15.5

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We wish to approximate a desired unit-sample response using the least-squares inverse design procedure as outlined in section 5.3.3. of text. The desired unit sample response hd(n) is given by

corresponding to a 9th order FIR filter. This is to be approximated by a second order IIR filter of the form

Solution 15.5

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The coefficient b0 is chosen to be hd(0) = 1. The coefficients a1 and a2 are obtained by solving the equations

For the specified desired unit-sample response