Convolution with numpy

A convolution is a way to combine two sequences, x and w, to get a third sequence, y, that is a filtered version of x. The convolution of the sample x_t is computed as follows:

It is the mean of the weighted summation over a window of length k and w_t are the weights. Usually, the sequence w is generated using a window function. Numpy has a number of window functions already implemented: bartlett, blackman, hamming, hanning and kaiser. So, let's plot some Kaiser windows varying the parameter beta:

import numpy
import pylab

beta = [2,4,16,32]

pylab.figure()
for b in beta:
 w = numpy.kaiser(101,b) 
 pylab.plot(range(len(w)),w,label="beta = "+str(b))
pylab.xlabel('n')
pylab.ylabel('W_K')
pylab.legend()
pylab.show()

The graph would appear as follows:

And now, we can use the function convolve(...) to compute the convolution between a vector x and one of the Kaiser window we have seen above:

def smooth(x,beta):
 """ kaiser window smoothing """
 window_len=11
 # extending the data at beginning and at the end
 # to apply the window at the borders
 s = numpy.r_[x[window_len-1:0:-1],x,x[-1:-window_len:-1]]
 w = numpy.kaiser(window_len,beta)
 y = numpy.convolve(w/w.sum(),s,mode='valid')
 return y[5:len(y)-5]

Let's test it on a random sequence:

# random data generation
y = numpy.random.random(100)*100 
for i in range(100):
 y[i]=y[i]+i**((150-i)/80.0) # modifies the trend

# smoothing the data
pylab.figure(1)
pylab.plot(y,'-k',label="original signal",alpha=.3)
for b in beta:
 yy = smooth(y,b) 
 pylab.plot(yy,label="filtered (beta = "+str(b)+")")
pylab.legend()
pylab.show()

The program would have an output similar to the following:

As we can see, the original sequence have been smoothed by the windows.

The sampling theorem explained with numpy

The sampling theorem states that a continuous signal x(t) bandlimited to B Hz can be recovered from its samples x[n] = x(n*T), where n is an integer, if T is greater than or equal to 1/(2B) without loss of any information. And we call 2B the Nyquist rate.
Sampling at a rate below the Nyquist rate is called undersampling, it leads to the aliasing effect. Let's observe the aliasing effect with the following script:

from numpy import linspace,cos,pi,ceil,floor,arange
from pylab import plot,show,axis

# sampling a signal badlimited to 40 Hz 
# with a sampling rate of 800 Hz
f = 40;  # Hz
tmin = -0.3;
tmax = 0.3;
t = linspace(tmin, tmax, 400);
x = cos(2*pi*t) + cos(2*pi*f*t); # signal sampling
plot(t, x)

# sampling the signal with a sampling rate of 80 Hz
# in this case, we are using the Nyquist rate.
T = 1/80.0;
nmin = ceil(tmin / T);
nmax = floor(tmax / T);
n = arange(nmin,nmax);
x1 = cos(2*pi*n*T) + cos(2*pi*f*n*T);
plot(n*T, x1, 'bo')

# sampling the signal with a sampling rate of 35 Hz
# note that 35 Hz is under the Nyquist rate.
T = 1/35.0;
nmin = ceil(tmin / T);
nmax = floor(tmax / T);
n = arange(nmin,nmax);
x2 = cos(2*pi*n*T) + cos(2*pi*f*n*T);
plot(n*T, x2, '-r.',markersize=8)

axis([-0.3, 0.3, -1.5, 2.3])
show()

The following figure is the result:

The blue curve is the original signal, the blue dots are the samples obtained with the Nyquist rate and the red dots are the samples obtainde with 35 Hz. It's easy to see that the blue samples are enough to recover the blue curve, while the red ones are not enough to capture the oscillations of the signal.

How to plot the frequency spectrum with scipy

Spectrum analysis is the process of determining the frequency domain representation of a time domain signal and most commonly employs the Fourier transform. The Discrete Fourier Transform (DFT) is used to determine the frequency content of signals and the Fast Fourier Transform (FFT) is an efficient method for calculating the DFT. Scipy implements FFT and in this post we will see a simple example of spectrum analysis:

from numpy import sin, linspace, pi
from pylab import plot, show, title, xlabel, ylabel, subplot
from scipy import fft, arange

def plotSpectrum(y,Fs):
 """
 Plots a Single-Sided Amplitude Spectrum of y(t)
 """
 n = len(y) # length of the signal
 k = arange(n)
 T = n/Fs
 frq = k/T # two sides frequency range
 frq = frq[range(n/2)] # one side frequency range

 Y = fft(y)/n # fft computing and normalization
 Y = Y[range(n/2)]
 
 plot(frq,abs(Y),'r') # plotting the spectrum
 xlabel('Freq (Hz)')
 ylabel('|Y(freq)|')

Fs = 150.0;  # sampling rate
Ts = 1.0/Fs; # sampling interval
t = arange(0,1,Ts) # time vector

ff = 5;   # frequency of the signal
y = sin(2*pi*ff*t)

subplot(2,1,1)
plot(t,y)
xlabel('Time')
ylabel('Amplitude')
subplot(2,1,2)
plotSpectrum(y,Fs)
show()

The program shows the following figure, on top we have a plot of the signal and on the bottom the frequency spectrum.