Basic operations on signal sequences – Addition

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Key focus: How to implement the basic addition operation on two discrete time signal sequences. Python code for signal addition is provided.

Signal addition

Given two discrete-time sequences \(x_1[n]\) and \(x_2[n]\), the addition of these two sequences is represented as \(x_1[n]+ x_2[n]\). The start of the sample at \(n=0\) can be different for these two sequences.

For example, if

\[ \begin{align} x_1[n] &= \left\{ -3, 12, -4, \underset{\uparrow}{9}, 2, 15\right\} \\ x_2[n] &= \left\{ -6, \underset{\uparrow}{4}, -2, 10\right\} \end{align} \]

the position of the sample at \(n=0\) is different in these sequences. Also, the length of these sequences are different.

The addition operation should take these differences into account. The following python code adjusts for these differences before computing the addition operation. It creates two sequences and of equal length that spans the minimum and maximum indices of and . The sequences and are first filled with zeros and then the sequences & are copied over to & at corresponding positions. The final result is computed as the sum of and .

import numpy as np

def signal_add(x1, n1, x2, n2):
    '''
    Computes y(n) = x1(n) + x2(n)
    ---------------------------------
    [y, n] = signal_add(x1, n1, x2, n2)
    
    Parameters
    ----------
    n1 = indices of first sequence x1
    n2 = indices of second sequence x2
    x1 = first sequence defined for indices n1
    x2 = second sequence defined for indices n2
    Returns
    -------
    y = output sequence defined over indices n that
    covers whole range of n1 and n2
    '''

    n_start = min(min(n1), min(n2))
    n_end = max(max(n1), max(n2))
    n = np.arange(n_start, n_end + 1)  # duration of y(n)

    y1 = np.zeros_like(n, dtype='complex_')
    y2 = np.zeros_like(n, dtype='complex_')

    mask1 = (n >= n1[0]) & (n <= n1[-1])
    mask2 = (n >= n2[0]) & (n <= n2[-1])

    y1[np.where(mask1)[0]] = x1[np.where(n1 == n[mask1])]
    y2[np.where(mask2)[0]] = x2[np.where(n2 == n[mask2])]

    y = y1 + y2

    return y, n

The following code snippet uses the function above to compute the addition of two sequences \(x_1[n]\) and \(x_2[n]\), defined as

\[\begin{align} x_1[n] &= cos \left( 0.03 \pi n \right), & 0 \leq n \leq 50 \\ x_2[n] &= e^{0.06 n}, & -30 \leq 0 \leq n \end{align}\]
n1 = np.arange(0,50)
n2 = np.arange(-30,10)
x1 = np.cos(0.03*np.pi*n1)
x2 = np.exp(0.06*n2)
[y,n] = signal_add(x1, n1, x2, n2)

These discrete sequences are plotted (Figure 1) using the code below.

import matplotlib.pyplot as plt
f, (ax1, ax2, ax3) = plt.subplots(3, 1, sharex=True, sharey=True)

ax1.stem(n1, x1, 'k', label = '$x_1[n]$')
ax2.stem(n2, x2, 'b', label = '$x_2[n]$')
ax3.stem(n, y, 'r', label = '$y[n] = x_1[n] + x_2[n]$')

ax1.legend(loc="upper right")
ax2.legend(loc="upper right")
ax3.legend(loc="upper right")

ax1.set_xlim((min(n), max(n)+1))
ax1.set_title('Addition of signals')
plt.show()
Addition of discrete signals in signals and systems. Signal addition
Figure 1: Addition of discrete signals

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