discrete convolve vs. analytical

conv.py

import numpy as np
import matplotlib.pyplot as plt


def convolve(y1, y2, dx=None):
    '''
    Compute the finite convolution of two signals of equal length.
    @param y1: First signal.
    @param y2: Second signal.
    @param dx: [optional] Integration step width.
    @note: Based on the algorithm at http://www.physics.rutgers.edu/~masud/computing/WPark_recipes_in_python.html.
    '''
    y1, y2 = map(np.asarray, [y1, y2])
    P = len(y1) #Determine the length of the signal
    z = np.zeros_like(y1) #Create a list of convolution values
    for k in range(P):
        t = 0
        lower = max(0, k - (P - 1))
        upper = min(P - 1, k)
        for i in range(lower, upper):
            t += (y1[i] * y2[k - i] + y1[i + 1] * y2[k - (i + 1)]) / 2
        z[k] = t
    if dx is not None: #Is a step width specified?
        z *= dx
    return z

steps = 50 #Number of integration steps
maxtime = 5 #Maximum time
dt = float(maxtime) / steps #Obtain the width of a time step

t = np.linspace(0, maxtime, steps, endpoint=False) #Create an array of times
exp1 = np.exp(-t) #Create an array of function values
exp2 = 2*np.exp(-2*t)

# Calculate the analytical expression
analytical = exp2*(np.exp(t)-1)
# Calculate the trapezoidal convolution
trapezoidal = convolve(exp1, exp2, dt)

# Calculate the scipy convolution
sci = np.convolve(exp1, exp2)
# Slice the first half to obtain the causal convolution and multiply
# by dt to account for the step width
sci = sci[:steps]*dt

# shift right, multiply
sci = np.r_[0,sci[:steps-1]]*0.86

# Plot
plt.plot(t, analytical, label = 'analytical')
plt.plot(t, trapezoidal, 'o', label = 'trapezoidal')
plt.plot(t, sci, '.', label = 'numpy.convolve')
plt.legend()
plt.show()