5.16 Digital signal processing problems  (Page 5/6)

Digital filtering

A digital filter has an input-output relationship expressed by the difference equation $$y(n)=\frac{x(n)+x(n-1)+x(n-2)+x(n-3)}{4}$$ .

1. Plot the magnitude and phase of this filter's transfer function.
2. What is this filter's output when $x(n)=\cos \left(\frac{\pi n}{2}\right)+2\sin \left(\frac{2\pi n}{3}\right)$ ?

Detective work

The signal $x(n)$ equals $\delta (n)-\delta (n-1)$ .

1. Find the length-8 DFT (discrete Fourier transform) of this signal.
2. You are told that when $x(n)$ served as the input to a linear FIR (finite impulse response) filter, the output was $y(n)=\delta (n)-\delta (n-1)+2\delta (n-2)$ . Is this statement true? If so, indicate why andfind the system's unit sample response; if not, show why not.

A discrete-time, shift invariant, linear system produces an output $y(n)=\{1, -1, 0, 0, \dots \}$ when its input $x(n)$ equals a unit sample.

1. Find the difference equation governing the system.
2. Find the output when $x(n)=\cos (2\pi f_{0}n)$ .
3. How would you describe this system's function?

Time reversal has uses

A discrete-time system has transfer function $H(e^{i\times 2\pi f})$ . A signal $x(n)$ is passed through this system to yield the signal $w(n)$ . The time-reversed signal $w(-n)$ is then passed through the system to yield the time-reversed output $y(-n)$ . What is the transfer function between $x(n)$ and $y(n)$ ?

Removing “hum”

The slang word “hum” represents power line waveforms that creep into signals because of poor circuitconstruction. Usually, the 60 Hz signal (and its harmonics) are added to the desired signal. What weseek are filters that can remove hum. In this problem, the signal and the accompanying hum have been sampled;we want to design a digital filter for hum removal.

1. Find filter coefficients for the length-3 FIR filter that can remove a sinusoid having digital frequency $f_0$ from its input.
2. Assuming the sampling rate is $f_s$ to what analog frequency does $f_0$ correspond?
3. A more general approach is to design a filter having a frequency response magnitude proportional to the absolute value of a cosine: $\left|H(e^{i\times 2\pi f})\right|\propto \left|\cos (\pi fN)\right|$ . In this way, not only can the fundamental but also its first few harmonics be removed. Select theparameter $N$ and the sampling rate so that the frequencies at which the cosine equals zerocorrespond to 60 Hz and its odd harmonics through the fifth.
4. Find the difference equation that defines this filter.

Digital am receiver

Thinking that digital implementations are always better, our clever engineer wants to design a digital AM receiver. The receiverwould bandpass the received signal, pass the result through an A/D converter, perform all the demodulationwith digital signal processing systems, and end with a D/A converter to produce the analog message signal.Assume in this problem that the carrier frequency is always a large even multiple of the message signal's bandwidth $W$ .

1. What is the smallest sampling rate that would be needed?
2. Show the block diagram of the least complex digital AM receiver.
3. Assuming the channel adds white noise and that a $b$ -bit A/D converter is used, what is the output's signal-to-noiseratio?

Dfts

A problem on Samantha's homework asks for the 8-point DFT of the discrete-time signal $\delta (n-1)+\delta (n-7)$ .