# Wave Walker DSP

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DSP for Beginners: Simple Explanations for Complex Numbers! The second edition includes a new chapter on complex sinusoids.

Calculate Average Signal Power
June 1, 2024

#### Introduction

Average signal power is important in calculating performance metrics such as signal to noise (SNR) ratio. Read on for how to derive and calculate the average signal power for a finite-length, discrete-time signal. Be sure to check out the post on deriving power and energy for a discrete-time signal.

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#### Signal Power Derivation

The average power of a signal x[n] is defined as

(1)

The expectation operator operates as an average or mean. For simplicity, and to make this practical for real-world RF systems, it is assumed that x[n] is N samples long beginning at time n=0. Therefore the expectation from (1) can be expanded according to

(2)

#### Examples of Signal Power with Math

Assume x[n] is a constant value A over N samples according to

(3)

The average power of signal x[n] is therefore

(4)

Assume y[n] is a complex sinusoid with frequency and amplitude A,

(5)

The average power is therefore

(6)

#### Calculating Power with Python

The average power for a complex sinusoid with with amplitude A = 3 is calculated according to (6),

(7)

Equations (6) and (7) are verified using simulation. The following code creates a complex sinusoid  of N=1024 samples with amplitude A = 3 and a random frequency and then computes the average power:

import numpy as np
# build a complex sinusoid
A = 3
omega = np.random.uniform(-np.pi,np.pi)
n = np.arange(0,1024)
complexSinusoid = A*np.exp(1j*omega*n)
# calculate average poweraveragePower = np.mean(np.abs(complexSinusoid)**2)print('avg power = ' + str(np.round(averagePower,2)))

Running the script produces:

avg power = 9.0

which aligns with the mathematical calculation.

#### Conclusion

This blog describes the average power mathematically, provides a derivation for the average power of a complex sinusoid and verifies the result using a Python simulation.

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