# Wireless 101: Peak to average power ratio (PAPR)

*This series is excerpted from "RF and Digital Signal Processing for Software-Defined Radio." Order this book today at www.elsevierdirect.com or by calling 1-800-545-2522 and receive an additional 20% discount and free shipping. Use promotion code 94446 when ordering. Valid only in North America.
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Part 3 lays out the basics of automatic gain control.*

**7.3.4 Peak to Average Power Ratio**

**7.3.4.1 PAR of a Single Tone Signal**

Define the peak to average power ratio (PAR) of a given signal *r*(*t*) as the ratio of the peak power of *r*(*t*) to its average power:

For a single tone *r*(*t*) = *A *cos (2p*F _{c }t*), the PAR can be estimated as:

The peak power of *r*(*t*) can be simply found as:

This is true obviously for the case where *t *= 2 *n*p, *n *= 0,1, … On the other hand, the average power of *r*(*t*) is found as:

The PAR of *r*(*t*) can then be estimated:

**7.3.4.2 PAR of Multi-Tone Signal**

Next we extend the results obtained for a single tone in the previous section to a signal that possesses multiple tones. To simplify the analysis, consider the two-tone signal case first.

Let the tones be harmonically related, that is *F*_{2} = *MF*_{1}. This is certainly the case of OFDM signals where the various subcarriers are separated by a fixed frequency offset. Consider:

The peak power of *r*(*t*) can be found by first looking at the square of (7.67):

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The peak power of *r*(*t*) can be found by taking the derivative of (7.68) and setting it equal to zero, that is:

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The relationship in (7.69) occurs for *t *= 2*np*, *n *= 0, 1, … , that is:

The absolute value in (7.70) was dropped since *r*(*t*) in this case is a real valued continuous function. Next, the average power of *r*(*t*) is found as:

*(Click to enlarge)*

The PAR is then the ratio:

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At this point it is instructive to look at two different cases. The first is when the two tones are of equal power, that is *A *= *B.* The relationship in (7.72) then becomes:

In (7.73), the result can be generalized to say that if the signal is made up of *N *tones that are of equal amplitude and that their phases are coherent in the sense that they add up constructively once over a certain period, then the PAR of this *N *-tone signal can be estimated as:

Note that, in reality, given an OFDM signal, for example, which is made up of *N *tones, the result presented in (7.74) is most unlikely to occur in the statistical sense due to the random nature of the data.

Next, we consider the case in which one tone is amplitude dominant over the other tone, say *A >> B. *In this case, the relationship in (7.72) can be approximated as:

*(Click to enlarge)*

The result in (7.75) implies that the PAR value is mostly influenced by the amplitude of the tones. If one of the tones is dominant, for example, then the resulting PAR is that of a single tone. In reality, in a multi-tone signal, the results are somewhere in between.