Sampling and reconstruction

This notebook will illustrate some effects of sampling and reconstruction

The sampling theorem

Shannon's sampling theorem says that a continuous-time signal with Fourier transform that is zero outside the interval $(-\omega_0, \omega_0)$ can be completely reconstructed from equidistant samples of the signal, as long as the sampling frequency is at least $2\omega_0$.



In [1]:

    
using PyPlot









    



INFO: Recompiling stale cache file /home/kjartan/.julia/lib/v0.6/PyCall.ji for module PyCall.
INFO: Recompiling stale cache file /home/kjartan/.julia/lib/v0.6/PyPlot.ji for module PyPlot.

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In [3]:

    
ω0 = 2;
x(t) = 2*sin.(ω0*t + pi/3);
t = linspace(0, 3, 700);
figure(figsize=(14,4));
pc = plot(t, x(t), linewidth=2);
xlabel("Seconds")

ωs = 20; # > 2ω0
h = 2*pi/ωs;
td = 0:h:3;
pd = plot(td, x(td), "o");
stem(td, x(td), linefmt="r--", markerfmt="ro", basefmt = "r-");
title("Continuous-time and sampled signal")









    












    Out[3]:





PyObject <matplotlib.text.Text object at 0x7f5987230610>

Reconstruction

Consider the continuous-time signal $x(t)$, which is sampled with sampling period $h$ and thus sampling frequency $$\omega_s = \frac{2\pi}{h}$$ to give the sampled signal $x[k] = x(kh)$. We assume that the signal $x(t)$ is bandlimited with frequency content zero above the frequency $\omega_0 < \omega_s/2$.

The signal can be reconstructed by the interpolation $$ \DeclareMathOperator{\sinc}{sinc} x(t) = \sum_{k=-\infty}^{\infty} x(kh) \sinc \frac{\omega_s(t-kh)}{2} = \sum_{k=-\infty}^{\infty} x(kh) \frac{ \sin \big(\omega_s(t-kh)/2\big)}{\omega_s(t-kh)/2} $$

The sinc function

The Fourier transform of the $\sinc$ function is the rectangular function, illustrated below.



In [9]:

    
# The sinc function and its Fourier transform
N = 300;
Nhalf = div(300,2)
ts = linspace(-3,3,N);
dt = (3+3)/N;
ωfft = 2π/dt;

figure(figsize=(14,3))
ps = plot(ts, sinc.(ωs*ts/2/π), linewidth=2); 
# Note that the sinc function in julia is defined as sin(πx)/(πx)

xlabel("seconds")

S = fft(sinc.(ωs*ts/2/π)); 
w = linspace(0, ωfft/2, Nhalf)
figure(figsize=(14,3))
pS = plot(w[1:Nhalf], abs.(S[1:Nhalf]), linewidth=2)
annotate(L"$\frac{\omega_s}{2}=10$ rad/s",
xy=[ωs/2;0],# Arrow tip
    xytext=[ωs/2+10;3], # Text offset from tip
size=16,
    xycoords="data", # Coordinates in in "data" units
arrowprops=Dict("facecolor"=>"black")) 
# Julia dictionary objects are automatically converted to Python object when they pass into a PyPlot function
#arrowprops=Dict("arrowstyle"=>"->", "facecolor"=>"black")) 

xticks([0, ωs/2, ωfft/2], ("0", L"$\omega_s/2$", "160"), fontsize="large");
xlabel("Frequency [rad/s]");

Exercise

Why is the Fourier transform plotted above not a perfect rectangular function?



In [19]:

    
?fft









    



search: fft fft! FFTW fftshift rfft ifft bfft ifft! bfft! ifftshift irfft brfft







    Out[19]:





fft(A [, dims])
Performs a multidimensional FFT of the array A.  The optional dims argument specifies an iterable subset of dimensions (e.g. an integer, range, tuple, or array) to transform along.  Most efficient if the size of A along the transformed dimensions is a product of small primes; see nextprod().  See also plan_fft() for even greater efficiency.
A one-dimensional FFT computes the one-dimensional discrete Fourier transform (DFT) as defined by
$\operatorname{DFT}(A)[k] =
  \sum_{n=1}^{\operatorname{length}(A)}
  \exp\left(-i\frac{2\pi
  (n-1)(k-1)}{\operatorname{length}(A)} \right) A[n].$
A multidimensional FFT simply performs this operation along each transformed dimension of A.
Higher performance is usually possible with multi-threading. Use FFTW.set_num_threads(np) to use np threads, if you have np processors.



In [36]:

    
?sinc









    



search: sinc significand isinteractive IntrinsicFunction DimensionMismatch







    Out[36]:





sinc(x)
Compute $\sin(\pi x) / (\pi x)$ if $x \neq 0$, and $1$ if $x = 0$.

New frequencies introduced by sample-and-hold

We have sampling frequency $\omega_s$ and Nyquist frequency $\omega_N = \frac{\omega_s}{2}$. A continous sinusoid signal of frequency $\omega_1 = 0.8\omega_N$ is sampled and then reconstructed using zero-order-hold. How does the spectrum look like?



In [60]:

    
ωs = 20
h = 2*pi/ωs
ωN = ωs/2.0
ω1 = 0.8ωN;
# Sampling instants
N = 1000
t = h*linspace(0,N-1, N)
xs = sin.(ω1*t)
# Construct zero-order-hold "continuous" signal
Nc = 100
hc = h/Nc
ωfft = 2π/hc;
x_zoh = repeat(xs, inner=Nc)
tc = hc*linspace(0, N*Nc-1, N*Nc)
figure(figsize=(14,3))
stem(t[1:100], xs[1:100], linefmt="r--", markerfmt="ro", basefmt = "r-")
plot(tc[1:10000], x_zoh[1:10000])
title("sampled signal and zero-order-hold")









    












    Out[60]:





PyObject <matplotlib.text.Text object at 0x7f5982e03090>



In [64]:

    
X_zoh = fft(x_zoh)
Nhalf = Int32(length(X_zoh)/2)
tTot = hc*length(x_zoh)
ωstep = 2*π/tTot
w = ωstep * linspace(0, Nhalf-1, Nhalf)
figure(figsize=(14,3))
    
pS = plot(w[1:1000], abs.(X_zoh[1:1000]), linewidth=2)









    












    Out[64]:





1-element Array{PyCall.PyObject,1}:
 PyObject <matplotlib.lines.Line2D object at 0x7f5982847f90>



In [52]:

    
0:ωstep:10









    Out[52]:





0.0:0.39999999999999997:10.0



In [51]:

    
w









    Out[51]:





0.0:0.4001600640256102:999.9999999999999



In [ ]: