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However, in order for yl(n) to be equal to yz(n), we require that y,(n) = wl(nM) = 0 This will be true if and only if M and L are relatively prime. n # kL
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Find two different continuous-time signals that will produce the sequence
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when sampled with a sampling frequency of 8 kHz. If the Nyquist rate for xu([) is R,, find the Nyquist rate for (a) x2(2t), (b) x(t/3), (c) x(t) * x(t). A continuous-time signal x,(t) is known to be uniquely recoverable from its samples xa(nTs) when T, = 1 ms. What is the highest frequency in Xu(f ) Suppose that x,(t) is bandlimited to 8 kHz (that is, Xu(f ) = 0 for (b) What is the Nyquist rate for x,(t)cos(2n . IOOOt)'
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8000). (a) What is the Nyquist rate for
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Let x,(t) = cos(6507ri) 2 sin(7207rt). (a) What is the Nyquist rate for x,(t) (b) If xa(t) is sampled at twice the Nyquist rate, what are the frequencies of the sinusoids in the sampled sequence If a continuous-time filter with an impulse response h,(r) is sampled with a sampling frequency off,, what happens to the cutoff frequency w,. of the discrete-time filter as 1,is increased
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A complex bandpass signal x,(r) with Xu(f') nonzero for 10 kHz < f < 12 kHz is sampled at a sampling rate of 2 kHz. The resulting sequence is
What is x,(t)
If the highest frequency in x,(t) is f = 8 kHz, find the minimum sampling frequency for the bandpass signal y,(t) = x,(OCOS(R~I) (a) Ro = 27r . 20 . 10' and (b) Ro = 217 . 24 . lo3. if
The continuous-time signal x,(t) = 7.25 cos(20007rt) is sampled at a sampling frequency of 8 kHz and quantized with a resolution A = 0.02. How many bits are required in the AID converter to avoid clipping xu([) Suppose that we want to sample the signal x,(t) with a 12-bit quantizer, where x,(t) is assumed to be gaussian with a variance u, . What is the signal-to-quantization noise ratio if we want the range of the quantizer to extend from -30, to 3ax Suppose that an analog waveform is sampled with a sampling frequency of 10 kHz and that xu([) contains a strong 60-Hz interference signal. If the only information in x,(r) of interest is in the frequency band above
CHAP. 31
SAMPLING
60 Hz, the interference may be eliminated with a discrete-time high-pass filter that has a frequency response of the form
What is the smallest cutoff frequency w, that may be used and still eliminate the 60-Hz interference
True or False: If x ( n ) has a discrete-time Fourier transform that is equal to zero for n / 4 < Iwl
I r,
Discrete-Time Processing of Analog Signals The system shown in Fig. 3-9 may be used to process an analog signal with a discrete-time system. Assume that x,(t) is bandlimited with X,( f ) = 0 for If 1 > 10 kHz as shown in the figure below.
If the discrete-time system is an ideal low-pass filter with a cutoff frequency of n/4, find the Fourier transform of y o ( [ )when ( a ) f i = 20 kHz and f 2 = 10 kHz and ( b )f l = 10 kHz and fi = 20 kHz. For bandlimited inpul signals, the system shown in Fig. 3-10 is a linear time-invariant continuous-time system. If
find the frequency response of the equivalent continuous-time system. For bandlimited input signals, the system shown in Fig. 3- 1 is a linear time-invariant continuous-time system. If 0 the overall system is to be a differentiator,
how should the frequency response of the discrete-time system be defined
Sample Rate Conversion
The up-sampler and down-sampler are components that are found in interpolators and decimators, respectively. Are these systems linear Are they shift-invariant A sequence x ( n ) corresponds to samples of a bandlimited signal using a sampling frequency of 10 kHz. However, the sequence should have been sampled using a sampling frequency f , = 12 kHz. Design a system for digitally changing the sampling rate.
A signal x,(r) that is bandlimited to 10 kHz is processed by the following system:
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