Page 200 - DSP Integrated Circuits

Problems 185

4.23 A digital signal can be characterized by the sum of two sinusoids:

where f\ = 0.8 kHz, /2 = 1.6 kHz, and fsample - 4 kHz. The sample rate is
decreased to 2 kHz by first applying a digital lowpass filter and then
discarding every other output sample.
(a) Determine first the digital spectrum if the lowpass filter is not used.
(b) Determine the digital spectrum if the lowpass filter meets the following
specification: A max = 0.1 dB, (O CT = n/4,A mi n = 40 dB, and Q) ST = 0.8 n.
4.24 The sample rate of the digital signal in Problem 4.23 is increased to 8 kHz by
inserting zeros and removing the unwanted images using a lowpass filter.
(a) Determine first the digital spectrum after zeros have been introduced.
(b) Determine proper cutoff and stopband angles for the lowpass filter.
(c) Determine the digital spectrum if the lowpass filter meets the following
specification A max = 0.1 dB andA mj ra = 40 dB.
(d) Determine the required order if a Cauer filter is used as the interpolator
filter.
4.25 (a) Determine the specification for an ideal interpolator that increases the
sampling frequency by a factor two for a bandlimited signal contained
within the band 0 to a^T < n.
(b) Determine the frequency response of the ideal interpolator filter.
(c) Determine the impulse response of the ideal interpolator filter.

4.26 Show that it is not efficient to interpolate the input sample rate by a factor
two by repeating the input samples instead of introducing zeros.
4.27 Suggest a system for interpolation of the sample frequency by a factor 1.75.
Determine also the cutoff and stopband edges of the filters involved.
4.28 The following scheme is used in a simple approach to double the sample
frequency. The original samples are retained and new samples are created
between the original samples by linear interpolation, i.e.,

(a) Determine the difference equation for the interpolator. Hint: Create a
new sequence from the original by interleaving zero samples between
the original samples.
(b) Determine the transfer function for the interpolator.
(c) Determine the poles and zeros.
(d) Sketch the frequency response.
(e) Sketch the Fourier transform for the original, input, and output signals
to the interpolator.
(f) Determine the impulse response of the interpolator.
4.29 Determine the work load required for the interpolator (case study) if the
allpass filter is placed after the interpolator filter.