SAMPLING

WHAT YOU ARE LOOKING AT

A digital computer cannot hold a continuous-time signal, so the top left panel is an approximation: a record of N points, sampled so fast that we can pretend it is continuous. Call that rate f_c. It carries a tone at f₁ and, unless you turn them off, two sidebands at f₁ ± Δf, plus a little Gaussian noise.

Sampling is a multiplication in the time domain by a pulse train: P samples long, the first few of them 1 and the rest 0. Multiplication in time is convolution in frequency, and the spectrum of a pulse train is a comb of lines at every multiple of f_s = f_c/P with a sinc envelope set by the duty cycle. So the sampled spectrum, bottom right, is the original spectrum copied to every multiple of f_s.

THINGS WORTH TRYING

• Raise f_1 bin past N/(2P). The tone crosses the Nyquist frequency f_s/2 and its image walks back down towards DC: that is aliasing, and after sampling you cannot tell the two apart. The alias readout tells you where it lands.
• Turn the Hann window off. With no window the FFT sees a record that does not contain a whole number of periods, the discontinuity at the wrap-around smears energy across every bin, and the floor rises far enough to bury the sidebands. This is why np.hanning is in every one of these scripts.
• Now tick coherent sampling as well. Every tone lands exactly on a bin, the record wraps around seamlessly, and the unwindowed spectrum is suddenly the cleanest of the four combinations. Coherent sampling is the alternative to windowing, and it is better when you can arrange it. ex/dt.py deliberately parks its tone halfway between two bins (f1 = 233.5/N), so its record is never coherent and the window always has a job to do.
• Push the duty cycle towards 1 (on phases = P). The images do not move, but the sinc envelope narrows and the high-frequency images fade: a long sampling phase is a low-pass filter on the images. Drop it to one phase in 64 and every image is nearly the same height.
• Set the sideband amplitude to 0 to get ex/sub.py's single-tone case, and drop the record length to 2¹⁰ to watch the noise floor rise: fewer FFT bins means the same noise power is shared between fewer of them.

THE PYTHON

#- Create the "continuous time" signal
f1 = 233/N
fd = 1/N*119
x_s = np.sin(2*np.pi*f1*t) + 1/1024*np.random.randn(N) \
    + 0.5*np.sin(2*np.pi*(f1-fd)*t) + 0.5*np.sin(2*np.pi*(f1+fd)*t)

#- Create the sampling vector, and the sampled signal
t_s_unit = [1,1,0,0,0,0,0,0]
t_s = np.tile(t_s_unit,int(N/len(t_s_unit)))
x_sn = x_s*t_s

#- Hanning window to avoid FFT bin energy spread
w = np.hanning(N+1)
X_s  = np.fft.fftshift(np.fft.fft(np.multiply(w[0:N],x_s)))
X_sn = np.fft.fftshift(np.fft.fft(np.multiply(w[0:N],x_sn)))

Source: ex/dt.py, ex/sub.py. The page uses the same formulas; the noise is seeded so the floor stays put while you drag a slider.