Sampling Exploration

I needed to review the Nyquist–Shannon sampling theorem. The coolest thing about Nyquist is that it expresses the sample-rate in terms of the function’s bandwidth and leads to a formula for the mathematically ideal interpolation algorithm.

What is sampling?

Sampling is nothing more than converting a signal into a numeric sequence. Shannon states:

If a function x(t) contains no frequencies higher than B hertz, it is completely determined by giving its ordinates at a series of points spaced 1/(2B) seconds apart.

This is really simple: A sufficient sample-rate is therefore samples/second, or anything larger. The two thresholds, and , are respectively called the Nyquist rate and Nyquist frequency.

I can’t understand something unless I explore with it. So, I defined the following code.

First we have to make the following definitions for a given sample-rate of :

• $$\scriptstyle T \stackrel{\mathrm{def}}{=}\ 1/f_s$$ represents the interval between samples.

Say we have a simple sign wave modulated at a frequency of 1/8.

If we explore, we can get:

Aliasing

When the bandlimit is too high (or there is no bandlimit), the reconstruction exhibits imperfections known as aliasing.

Let be the Fourier transform of bandlimited function :

$$X(f) \stackrel{\mathrm{def}}{=}\ \int_{-\infty}^{\infty} x(t) \ e^{- i 2 \pi f t} \ {\rm d}t$$

and

$$X(f) = 0 \quad \text{for all} |f| > B$$

The Poisson summation formula shows that the samples, x(nT), of function x(t) are sufficient to create a periodic summation of function X(f). The result is:

$$X_s(f)\ \stackrel{\mathrm{def}}{=} \sum_{k=-\infty}^{\infty} X\left(f – k f_s\right) = \sum_{n=-\infty}^{\infty} \underbrace{T\cdot x(nT)}_{x[n]}\ e^{-i 2\pi n T f},$$

This function is also known as the discrete-time Fourier transform.

Questions I’m after:

• How is it that the samples of several different sine waves can be identical, when at least one of them is at a frequency above half the sample rate?

You can see my source code here:

	close all;
	clear all;
	t=-10:0.01:10;
	T=8;
	fm=1/T;
	x=cos(2*pi*fm*t);
	fs1=1.2*fm;
	fs2=2*fm;
	fs3=8*fm;
	n1=-4:1:4;
	xn1=cos(2*pi*n1*fm/fs1);
	subplot(221)
	plot(t,x);
	xlabel('time in seconds');
	ylabel('x(t)');
	title('continous time signal');
	subplot(222)
	stem(n1,xn1);
	hold on;
	plot(n1,xn1);
	xlabel('n');
	ylabel('x(n)');
	title('discrete time signal with fs<2fm');
	%
	n2=-5:1:5;
	xn2=cos(2*pi*n2*fm/fs2);
	subplot(223)
	stem(n2,xn2);
	hold on;
	plot(n2,xn2);
	xlabel('n');
	ylabel('x(n)');
	title('discrete time signal with fs=2fm');
	%
	n3=-20:1:20;
	xn3=cos(2*pi*n3*fm/fs3);
	subplot(224)
	stem(n3,xn3);
	hold on;
	plot(n3,xn3);
	xlabel('n');
	ylabel('x(n)');
	title('discrete time signal with fs>2fm');

view raw sampling.m hosted with by GitHub