Sampling and Filtering

The signal st s t is bandlimited to 4 kHz. We want to sample it, but it has been subjected to various signal processing manipulations.

Non-Standard Sampling

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

Figure 1

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 (Figure 2).

Figure 2

Bandpass Sampling

The signal st s t has the indicated spectrum.

Figure 3

Sampling Signals

If a signal is bandlimited to WW Hz, we can sample it at any rate 1Ts>2W 1 Ts 2 W and recover the waveform exactly. This statement of the Sampling Theorem can be taken to mean that all information about the original signal can be extracted from the samples. While true in principle, you do have to be careful how you do so. In addition to the rms value of a signal, an important aspect of a signal is its peak value, which equals max{|st|} s t .

Hardware Error

An A/D converter has a curious hardware problem: Every other sampling pulse is half its normal amplitude (Figure 4).

Figure 4

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 BB-bit converter. Thus, we want to convert numbers having a BB-bit representation into a voltage proportional to that number. The first step taken by our simple converter is to represent the number by a sequence of BB pulses occurring at multiples of a time interval TT. The presence of a pulse indicates a “1” in the corresponding bit position, and pulse absence means a “0” occurred. For a 4-bit converter, the number 13 has the binary representation 1101 ( 13 10 =1×23+1×22+0×21+1×20 13 10 1 2 3 1 2 2 0 2 1 1 2 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.

Figure 5

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

Discrete-Time Fourier Transforms

Find the Fourier transforms of the following sequences, where sn s n is some sequence having Fourier transform Sⅇⅈ2πf S 2 f .

Spectra of Finite-Duration Signals

Find the indicated spectra for the following signals.

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 sat=sin4000t sa t 4000t . To analyze the spectrum, he samples his recorded whistle with a sampling interval of TS=2.5×10-4 TS 2.5 10-4 to obtain sn=sanTS sn sa n TS . Sammy (wisely) decides to analyze a few samples at a time, so he grabs 30 consecutive, but arbitrarily chosen, samples. He calls this sequence xn xn and realizes he can write it as xn=sin4000nTS+θ ,   n=0…29 xn 4000n TS θ ,   n 0 … 29

Discrete-Time Filtering

We can find the input-output relation for a discrete-time filter much more easily than for analog filters. The key idea is that a sequence can be written as a weighted linear combination of unit samples.

A Digital Filter

A digital filter has the depicted unit-sample reponse.

Figure 6

A Special Discrete-Time Filter

Consider a FIR filter governed by the difference equation yn=13xn+2+23xn+1+xn+23xn−1+13xn−2 y n 13 x n 2 23 x n 1 x n 23 x n 1 13 x n 2

Simulating the Real World

Much of physics is governed by differntial equations, and we want to use signal processing methods to simulate physical problems. The idea is to replace the derivative with a discrete-time approximation and solve the resulting differential equation. For example, suppose we have the differential equation ddtyt+ayt=xt t y t a y t x t and we approximate the derivative by ddtyt|t=nT≈ynT−yn−1TT t n T t y t y n T y n 1 T T where TT essentially amounts to a sampling interval.

Derivatives

The derivative of a sequence makes little sense, but still, we can approximate it. The digital filter described by the difference equation yn=xn−xn−1 yn xn x n 1 resembles the derivative formula. We want to explore how well it works.

The DFT

Let's explore the DFT and its properties.

The Fast Fourier Transform

Just to determine how fast the FFT algorithm really is, we can take advantage of MATLAB's fft function. If x is a length-NN vector, fft(x) computes the length-NN transform using the most efficient algorithm it can. In other words, it does not automatically zero-pad the sequence and it will use the FFT algorithm if the length is a power of two. Let's count the number of arithmetic operations the fft program requires for lengths ranging from 2 to 1024.

		
		for n=2:1024,
		  x = randn(1,n);
		  t_start = cputime;
		  fft(x);
		  time(n) = cputime - t_start;
		end
		
		

DSP Tricks

Sammy is faced with computing lots of discrete Fourier transforms. He will, or course, use the FFT algorithm, but he is behind schedule and needs to get his results as quickly as possible. He gets the idea of computing two transforms at one time by computing the transform of sn=s1n+ⅈs2n s n s1 n s2 n , where s1n s1 n and s2n s2 n are two real-valued signals of which he needs to compute the spectra. The issue is whether he can retrieve the individual DFTs from the result or not.

Discrete Cosine Transform (DCT)

The discrete cosine transform of a length-NN sequence is defined to be Sck=∑n=0N−1sncos2πnk2N Sc k n 0 N 1 sn 2nk 2N Note that the number of frequency terms is 2N−1 2N 1 : k=0…2N−1 k 0 … 2N 1 .

A Digital Filter

A digital filter is described by the following difference equation: yn=ayn−1+axn−xn−1 , a=12 yn a y n 1 a xn x n 1 , a 1 2

Another Digital Filter

A digital filter is determined by the following difference equation. yn=yn−1+xn−xn−4 yn y n 1 x n x n 4

Yet Another Digital Filter

A filter has an input-output relationship given by the difference equation yn=14xn+12xn−1+14xn−2 y n 14 x n 12 x n 1 14 x n 2 .

Figure 8

A Digital Filter in the Frequency Domain

We have a filter with the transfer function Hⅇⅈ2πf=ⅇ-ⅈ2πfcos2πf H 2 f 2 f 2 f operating on the input signal xn=δn−δn−2 xn δn δ n 2 that yields the output yn yn.

Digital Filters

A discrete-time system is governed by the difference equation yn=yn−1+xn+xn−12 y n y n 1 x n x n 1 2

Digital Filtering

A digital filter has an input-output relationship expressed by the difference equation yn=xn+xn−1+xn−2+xn−34 y n x n x n 1 x n 2 x n 3 4 .

Detective Work

The signal xn x n equals δn−δn−1 δ n δ n 1 .

A discrete-time, shift invariant, linear system produces an output yn=1-100… y n 1 -1 0 0 … when its input xn x n equals a unit sample.

Time Reversal has Uses

A discrete-time system has transfer function Hⅇⅈ2πf H 2 f . A signal xn x n is passed through this system to yield the signal wn w n . The time-reversed signal w-n w n is then passed through the system to yield the time-reversed output y-n y n . What is the transfer function between xn x n and yn y n ?

Removing “Hum”

The slang word “hum” represents power line waveforms that creep into signals because of poor circuit construction. Usually, the 60 Hz signal (and its harmonics) are added to the desired signal. What we seek 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.

Digital AM Receiver

Thinking that digital implementations are always better, our clever engineer wants to design a digital AM receiver. The receiver would bandpass the received signal, pass the result through an A/D converter, perform all the demodulation with 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 W.

DFTs

A problem on Samantha's homework asks for the 8-point DFT of the discrete-time signal δn−1+δn−7 δ n 1 δ n 7 .

Stock Market Data Processing

Because a trading week lasts five days, stock markets frequently compute running averages each day over the previous five trading days to smooth price fluctuations. The technical stock analyst at the Buy-Lo--Sell-Hi brokerage firm has heard that FFT filtering techniques work better than any others (in terms of producing more accurate averages).

Echoes

Echoes not only occur in canyons, but also in auditoriums and telephone circuits. In one situation where the echoed signal has been sampled, the input signal xn xn emerges as xn+a1xn−n1+a2xn−n2 xn a1 x n n1 a2 x n n2 .

Digital Filtering of Analog Signals

RU Electronics wants to develop a filter that would be used in analog applications, but that is implemented digitally. The filter is to operate on signals that have a 10 kHz bandwidth, and will serve as a lowpass filter.

Signal Compression

Because of the slowness of the Internet, lossy signal compression becomes important if you want signals to be received quickly. An enterprising 241 student has proposed a scheme based on frequency-domain processing. First of all, he would section the signal into length-NN blocks, and compute its NN-point DFT. He then would discard (zero the spectrum) at half of the frequencies, quantize them to bb-bits, and send these over the network. The receiver would assemble the transmitted spectrum and compute the inverse DFT, thus reconstituting an NN-point block.