5.16 Digital signal processing problems  (Page 2/6)

Hardware error

An A/D converter has a curious hardware problem: Every other sampling pulse is half its normal amplitude( [link] ).

1. Find the Fourier series for this signal.
2. Can this signal be used to sample a bandlimited signal having highest frequency $W=\frac{1}{2T}$ ?

Simple d/a converter

Commercial digital-to-analog converters don't work this way, but a simple circuit illustrates how they work.Let's assume we have a $B$ -bit converter. Thus, we want to convert numbers having a $B$ -bit representation into a voltage proportional to that number. The first steptaken by our simple converter is to represent the number by a sequence of $B$ pulses occurring at multiples of a time interval $T$ . The presence of a pulse indicates a “1” in the correspondingbit position, and pulse absence means a “0” occurred. For a 4-bit converter, the number 13 has thebinary representation 1101 ( ${13}_{10}=12^3+12^2+02^1+12^0$ ) and would be represented by the depicted pulse sequence. Note that the pulse sequence is“backwards” from the binary representation. We'll see why that is.

This signal serves as the input to a first-order RC lowpass filter. We want to design the filter and the parameters $\Delta$ and $T$ so that the output voltage at time $4T$ (for a 4-bit converter) is proportional to the number. This combination of pulse creation andfiltering constitutes our simple D/A converter. The requirements are

• The voltage at time $t=4T$ should diminish by a factor of 2 the further the pulse occurs from this time. In other words, thevoltage due to a pulse at $3T$ should be twice that of a pulse produced at $2T$ , which in turn is twice that of a pulse at $T$ , etc.
• The 4-bit D/A converter must support a 10 kHz samplingrate.

Discrete-time fourier transforms

Find the Fourier transforms of the following sequences, where $s(n)$ is some sequence having Fourier transform $S(e^{i\times 2\pi f})$ .

1. $-1^{n}s(n)$
2. $s(n)\cos (2\pi f_{0}n)$
3. $x(n)=\begin{cases}s(\frac{n}{2}) & \text{if $n(\text{even})$}\\ 0 & \text{if $n(\text{odd})$}\end{cases}$
4. $ns(n)$

Spectra of finite-duration signals

Find the indicated spectra for the following signals.

1. The discrete-time Fourier transform of $s(n)=\begin{cases}\cos (\frac{\pi }{4}n)^2 & \text{if $n=\{-1, 0, 1\}$}\\ 0 & \text{if $\text{otherwise}$}\end{cases}$
2. The discrete-time Fourier transform of $s(n)=\begin{cases}n & \text{if $n=\{-2, -1, 0, 1, 2\}$}\\ 0 & \text{if $\text{otherwise}$}\end{cases}$
3. The discrete-time Fourier transform of $s(n)=\begin{cases}\sin (\frac{\pi }{4}n) & \text{if $n=\{0, \dots , 7\}$}\\ 0 & \text{if $\text{otherwise}$}\end{cases}$
4. The length-8 DFT of the previous signal.

Just whistlin'

Sammy loves to whistle and decides to record and analyze his whistling in lab. He is a very good whistler; his whistle is a pure sinusoid that can be described by $s_{\mathrm{a}}(t)=\sin (4000t)$ . To analyze the spectrum, he samples his recorded whistle with a sampling interval of $T_{\mathrm{S}}=2.510^{-4}$ to obtain $s(n)=s_{\mathrm{a}}(n{T}_{\mathrm{S}})$ . Sammy (wisely) decides to analyze a few samples at a time, so he grabs 30 consecutive, but arbitrarily chosen, samples.He calls this sequence $x(n)$ and realizes he can write it as $$x(n)=\sin (4000n{T}_{\mathrm{S}}+\theta )\text{, }n=\{0, \dots , 29\}$$

1. Did Sammy under- or over-sample his whistle?
2. What is the discrete-time Fourier transform of $x(n)$ and how does it depend on $\theta$ ?
3. How does the 32-point DFT of $x(n)$ depend on $\theta$ ?