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Calculate Average Signal Power
June 1, 2024

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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)   \begin{equation*}P_{x} = \mathbb{E} \{ x[n]  x^*[n] \}.\end{equation*}

The expectation operator \mathbb{E} \{ \cdot \} 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)   \begin{equation*}\begin{split}P_{x} & = \mathbb{E} \{ x[n] x^*[n] \} \\& = \frac{1}{N} \sum_{n=0}^{N-1} x[n] x^*[n]  \\& = \frac{1}{N} \sum_{n=0}^{N-1} |x[n]|^2.\end{split}\end{equation*}

Examples of Signal Power with Math

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

(3)   \begin{equation*}x[n] = \begin{cases}A, & 0 \le n < N-1 \\0, & \text{otherwise}.\end{equation*}

The average power of signal x[n] is therefore 

(4)   \begin{equation*}\begin{split}P_{x} & = \frac{1}{N} \sum_{n=0}^{N-1} A \cdot A \\& = \frac{1}{N} \sum_{n=0}^{N-1} A^2 \\& = \frac{1}{N} \cdot N \cdot A^2 \\& = A^2.\end{split}\end{equation*}

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

(5)   \begin{equation*}y[n] = \begin{cases}A e^{j\omega n}, & 0 \le n < N-1 \\0, & \text{otherwise}.\end{equation*}

The average power is therefore 

(6)   \begin{equation*}\begin{split}P_{y} & = \frac{1}{N} \sum_{n=0}^{N-1} A e^{j\omega n} A e^{-j\omega n} \\& = \frac{1}{N} \sum_{n=0}^{N-1} A^2 e^{j \omega n - j \omega n} \\& = \frac{1}{N} \cdot N \cdot A^2 \\& = A^2.\end{split}\end{equation*}

Calculating Power with Python

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

(7)   \begin{equation*}|A|^2 = |3|^2 = 9.\end{equation*}

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 \omega 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 power
averagePower = 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.


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.

More DSP blogs:

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