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Why doesn't matlab give me an 8KHz sinewave for 16KHz sampling frequency?

I have the following matlab code, and I am trying to get 64 samples of various sinewave frequencies at 16KHz sampling frequency: 

close all; clear; clc;
dt=1/16000;
freq = 8000;
t=-dt;
for i=1:64,
t=t+dt;a(i)=sin(2*pi*freq*t);
end
plot(a,'-o'); grid on;

for freq = 1000, the output graph is
enter image description here

The graph seems normal upto 2000, but at 3000, the graph is
enter image description here

We can see that the amplitude changes during every cycle

Again, at 4000 the graph is
enter image description here

Not exactly a sinewave, but the amplitude is as expected during every cycle and if I play it out it sounds like a single frequency tone

But again at 6000 we have
enter image description here

and at 8000 we have
enter image description here

Since the sampling frequency is 16000 I was assuming that I should be able to generate sinewave samples for upto 8000, and I was expecting the graph I got at 4000 to appear at 8000. Instead, even at 3000, the graph starts to look weird

If I change the sampling frequency to 32000 and the sinewave frequency to 16000, I get the same graph that I am getting now at 8000. Why does matlab behave this way?

EDIT:

at freq = 7900
enter image description here

Solution

  • This is just an artifact of aliasing. Notice how the vertical axis for the 8kHz graph only goes up to 1.5E-13? Ideally the graph should be all zeros; what you're seeing is rounding error.

    Looking at the expression for computing the samples at 16kHz:

    x(n) = sin(2 * pi * freq * n / 16000)

    Where x is the signal, n is the integer sample number, and freq is the frequency in hertz. So, when freq is 8kHz, it's equivalent to:

    x(n) = sin(2 * pi * 8000 * n / 16000) = sin(pi * n)

    Because n is an integer, sin(pi * n) will always be zero. 8kHz is called the Nyquist frequency for a sampling rate of 16kHz for this reason; in general, the Nyquist frequency is always half the sample frequency.

    At 3kHz, the signal "looks weird" because some of the peaks are at non-integer multiples of 16kHz, because 16 is not evenly divisible by 3. Same goes for the 6kHz signal.

    The reason they still sound like pure sine tones is because of how the amplitude is interpolated between samples. The graph uses simple linear interpolation, which gives the impression of harsh edges at the samples. However, a physical loudspeaker (more precisely, the circuitry which drives it) does not use linear interpolation directly. Instead, a small filter circuit is used to smooth out those harsh edges (aka anti-aliasing) which removes the artificial frequencies above the aforementioned Nyquist frequency.