5.16 Digital signal processing problems

Sampling and filtering

The signal $s(t)$ is bandlimited to 4 kHz. We want to sample it, but it has been subjected to various signal processingmanipulations.

1. What sampling frequency (if any works) can be used to sample the result of passing $s(t)$ through an RC highpass filter with $R=10\mathrm{k\Omega }$ and $C=8\mathrm{nF}$ ?
2. What sampling frequency (if any works) can be used to sample the derivative of $s(t)$ ?
3. The signal $s(t)$ has been modulated by an 8 kHz sinusoid having an unknown phase: the resultingsignal is $s(t)\sin (2\pi f_{0}t+\phi )$ , with $f_0=8\mathrm{kHz}$ and $\phi =?$ Can the modulated signal be sampled so that the original signal can be recovered from the modulated signal regardless of the phase value $\phi$ ? If so, show how and find the smallest sampling rate that can be used; if not,show why not.

Non-standard sampling

Using the properties of the Fourier series can ease finding a signal's spectrum.

1. Suppose a signal $s(t)$ is periodic with period $T$ . If $c_k()$ represents the signal's Fourier series coefficients, what are the Fourier seriescoefficients of $s(t-\frac{T}{2})$ ?
2. Find the Fourier series of the signal $p(t)$ shown in [link] .
3. Suppose this signal is used to sample a signal bandlimited to $\frac{1}{T}\mathrm{Hz}$ . Find an expression for and sketch the spectrum of the sampled signal.
4. Does aliasing occur? If so, can a change in sampling rate prevent aliasing;if not, show how the signal can be recovered from these samples.

Pulse signal

A different sampling scheme

A signal processing engineer from Texas A&M claims to have developed an improved sampling scheme. He multiplies the bandlimited signal by the depicted periodic pulse signal to perform sampling ( [link] ).

1. Find the Fourier spectrum of this signal.
2. Will this scheme work? If so, how should $T_S$ be related to the signal's bandwidth? If not, why not?

Bandpass sampling

The signal $s(t)$ has the indicated spectrum.

1. What is the minimum sampling rate for this signal suggested by the Sampling Theorem?
2. Because of the particular structure of this spectrum, one wonders whether a lower sampling ratecould be used. Show that this is indeed the case, and find the system that reconstructs $s(t)$ from its samples.

Sampling signals

If a signal is bandlimited to $W$ Hz, we can sample it at any rate $\frac{1}{T_s}> 2W$ and recover the waveform exactly. This statement of the Sampling Theorem can be taken to mean that allinformation about the original signal can be extracted from the samples. While true in principle, you do haveto be careful how you do so. In addition to the rms value of a signal, an important aspect of a signal isits peak value, which equals $\max\{\left|s(t)\right|\}$ .

1. Let $s(t)$ be a sinusoid having frequency $W$  Hz. If we sample it at precisely the Nyquist rate, how accurately do thesamples convey the sinusoid's amplitude? In other words, find the worst case example.
2. How fast would you need to sample for the amplitude estimate to be within 5% of the truevalue?
3. Another issue in sampling is the inherent amplitude quantization produced by A/D converters. Assume themaximum voltage allowed by the converter is $V_{\max }()$ volts and that it quantizes amplitudes to $b$ bits. We can express the quantized sample $Q(s(n{T}_s))$ as $s(n{T}_s)+\epsilon (t)$ , where $\epsilon (t)$ represents the quantization error at the $n^{\mathrm{th}}()$ sample. Assuming the converter rounds, how large is maximum quantization error?
4. We can describe the quantization error as noise, with apower proportional to the square of the maximum error. What is the signal-to-noise ratio of thequantization error for a full-range sinusoid? Express your result in decibels.