What is the practical meaning of variance, covariance and mean for a signal?


2 Answers 2


Adding to Zeeshan's useful answer, here are some additional comments toward some their practical use (not limited to this but may helps add intuition into their use depending on your background):

Variance: The average "AC" power quantity of a signal is directly proportional to the variance (simply the average of the squares; this relation to "power" is clear when considering electrical signals of current or voltage). The total power is due to the DC component and AC component, and the DC component as Zeeshan has shown is the mean. By definition of variance, the DC component is subtracted from the signal prior to taking the average of the squares, and thus is the power of the "AC" component only. Often in communications our concern is with AC as that is where the information content is (if I only send a "1" I am not really transmitting information beyond that).

The square root of the variance is the "standard deviation". (The "rms" of a signal is the standard deviation if the DC component is removed). Notice that the power due to a DC signal at level X is the same as an AC signal with rms (standard deviation) level X. Thus for AC signals, including noisy signals with no DC component, the standard deviation is often the magnitude quantity for that signal (and works out as such in many equations).

Covariance: Instead of taking the average of the squared terms for a single variance, we take the average of a variable multiplied by another variable, and thus covariance is dependent on two variables as is $cov(X,Y)$. Note that $cov(X,X)$ is identical to the variance. Co-variance is very closely related to correlation, which tells us the linear relationship between two signals. Often in signal processing we simply refer to the process of multiply and integrate (or in discrete signal processing: multiply and accumulate) for two signals as "correlation":

$$Cor = \int f(x)g^*(x)$$

$$Cor = \sum f[n]g^*[n]$$

(where (*) is the complex conjugate for complex signals).

It is interesting how prolific the above operations appear dominantly signal processing (Fourier transform, Laplace transform, Z transform, Viterbi Decoding, Matched Filter, down-conversion, spread spectrum demodulation, expected value, dot product, FIR filtering, etc) and this is because of the "processing gain" in SNR that the operation gives to signal components that are alike between the two variables.

If you ensure the DC component is removed from each signal, the computation of "correlation" as I presented is closely related to the covariance (just without the normalization in the final step of taking the average). If you divide the covariance by the standard deviation of each signal, the result is the normalized correlation coefficient, which will always be between +1 and -1. (Note my earlier comment that the standard deviation is the "effective magnitude" so that helps see how this would normalize the signal). Thus we see while the correlation coefficient is normalized to always be +/-1, the covariance will grow with the "cross-power" term of the two signals. Other than that they both show the degree of (linear) similarity between two variables.

Here is a practical application where the correlation coefficient was suggested for use: Noise detection

  • 1
    $\begingroup$ +1 for brief extension of the related topics. This extension in conjunction with above answer, provides the complete insight. $\endgroup$
    – Zeeshan
    Mar 29, 2017 at 12:47

Mean of a signal can be practically visualized as the dc average value present in the signal (for a complete sinusoidal period), for e.g

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Variance of a signal is the difference between the normalized squared sum of instantaneous values with the mean value. In other words it provides you with the deviation of the signal from its mean value. It gives you the spread of your signal's data set.

Co-variance of a signal consists of unnormalized relation with the signal correlated with itself and also with other signals. It provides you the degree of similarity between the signals of interest.






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