Skip to content Skip to navigation

Connexions

You are here: Home » Content » Signal Power

Navigation

Content Actions
  • Download module PDF
  • Add to ...
    Add the module to:
    • My Favorites
    • A lens
    • An external social bookmarking service
    • My Favorites (What is 'My Favorites'?)
      'My Favorites' is a special kind of lens which you can use to bookmark modules and collections directly in Connexions. 'My Favorites' can only be seen by you, and collections saved in 'My Favorites' can remember the last module you were on. You need a Connexions account to use 'My Favorites'.
    • A lens (What is a lens?)

      Definition of a lens

      Lenses

      A lens is a custom view of Connexions content. You can think of it as a fancy kind of list that will let you see Connexions through the eyes of organizations and people you trust.

      What is in a lens?

      Lens makers point to Connexions materials (modules and collections), creating a guide that includes their own comments and descriptive tags about the content.

      Who can create a lens?

      Any individual Connexions member, a community, or a respected organization.

      What are tags? tag icon

      Tags are descriptors added by lens makers to help label content, attaching a vocabulary that is meaningful in the context of the lens.

    • External bookmarks
  • E-mail the author
  • Rate this module (How does the rating system work?)

    Rating system

    Ratings

    Ratings allow you to judge the quality of modules. If other users have ranked the module then its average rating is displayed below. Ratings are calculated on a scale from one star (Poor) to five stars (Excellent).

    How to rate a module

    Hover over the star that corresponds to the rating you wish to assign. Click on the star to add your rating. Your rating should be based on the quality of the content. You must have an account and be logged in to rate content.

    		(0 ratings)

Recently Viewed

This feature requires Javascript to be enabled.

Signal Power

Module by: Don Johnson

Summary: This module examines signal power, looking at instantaneous and average power. It uses orthogonality properties to derive a simple expression for average power. It also defines and displays a power spectrum.

Note: Your browser may not currently support MathML. See our browser support page for additional details. You can always view the correct math in the PDF version.

An interesting question you could ask about a signal is its average power. A signal's instantaneous power is defined to be its square, as if it were a voltage or current passing through a 1 Ω resistor. The average power is the average of the instantaneous power over some time interval. For a periodic signal, the natural time interval is clearly its period; for nonperiodic signals, a better choice would be entire time or time from onset. For a periodic signal, the average power is the square of the root-mean-squared (rms) value. We define the rms value of a periodic signal to be

rmss=1T∫0Ts2tdt rms s 2 1 T t 0 T s t 2 (1)
and thus its average power is rms2s rms s 2 .
powers=rms2s=1T∫0Ts2tdt power s rms s 2 1 T t 0 T s t 2 (2)

Exercise 1

What is the rms value of the half-wave rectified sinusoid?

Solution

A half-wave rectified sinusoid has half the average power of the original sine wave since it is zero half the time. A sine wave's average power equals A22 A 2 2 , making the rms value of the half-wave rectified signal A2 A 2 .

To find the average power in the frequency domain, we need to substitute the spectral representation of the signal into this expression.

powers=1T∫0T a 0 +∑k=1∞ a k cos2πktT+∑k=1∞ b k sin2πktT2dt power s 1 T t 0 T a 0 k 1 a k 2 k t T k 1 b k 2 k t T 2 (3)
The square inside the integral will contain all possible pairwise products. However, the orthogonality properties say that most of these crossterms integrate to zero. The survivors leave a rather simple expression for the power we seek.
powers= a 0 2+12∑k=1∞ a k 2+ b k 2 power s a 0 2 1 2 k 1 a k 2 b k 2 (4)

Figure 1: Power spectrum of a half-wave rectified sinusoid.
Power Spectrum of a Half-Wave Rectified Sinusoid
Power Spectrum of a Half-Wave Rectified Sinusoid (fourier3.png)

It could well be that computing this sum is easier than integrating the signal's square. Furthermore, the contribution of each term in the Fourier series toward representing the signal can be measured by its contribution to the signal's average power. Thus, the power contained in a signal at its k k th harmonic is a k 2+ b k 22 a k 2 b k 2 2 . The power spectrum P s k P s k , such as shown in Figure 1, plots each harmonic's contribution to the total power.

Exercise 2

In stereophonic systems, deviation of a sine wave from the ideal is measured by the total harmonic distortion, which equals the total power in the harmonics higher than the first compared to power in the fundamental. Find an expression for the total harmonic distortion for any periodic signal. Is this calculation most easily performed in the time or frequency domain?

Solution

Total harmonic distortion equals ∑k=2∞ a k 2+ b k 2 a 1 2+ b 1 2 k 2 a k 2 b k 2 a 1 2 b 1 2 . Clearly, this quantity is most easily computed in the frequency domain. However, the numerator equals the the square of the signal's rms value minus the power in the average and the power in the first harmonic.

Comments, questions, feedback, criticisms?

Send feedback