DSP Tricks_ Frequency Demodulation Algorithms

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DSP Tricks: Frequency demodulation

algorithms

Richard Lyons - January 10, 2011

An often used technique for measurement of the instantaneous frequency of a

complex sinusoidal signal is to compute the derivative of the signals

instantaneous (n) phase as shown in Figure 1360 below.

Figure 1360 Frequency demodulator using an arctangent function.

This is the traditional discrete signal FM demodulation method, and it works

fine. The demodulators instantaneous output frequency is Eq13-111 below:

where f is the sample rate in Hz.

Computing instantaneous phase (n) requires an arctangent operation, which

is difficult to implement accurately without considerable computational

resources. Heres a scheme for computing (n) for use in Eq. (13111)

without the intermediate (n) phase computation (and its pesky arctan-gent).

We derive the (n) computation algorithm as follows, initially using

continuous-time variables based on the following definitions:

i(t) = in-phase signal,q(t) = quadrature phase signal,

(t) = tan [q(t)/i(t)] = instantaneous phase,

(t) = time derivative of (t). (13-112)

First, we let r(t)= q(t)/i(t) be the signal for which were trying to compute the

derivative of its arctangent. The time derivative of tan [r(t)], a calculus iden-

tity, is

Because d[r(t)]/dt = d[q(t)/i(t)]/dt, we use the calculus identity for the deriv-

ative of a ratio to write

s

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Plugging Eq. (13114)s result into Eq. (13113), we have

Replacing r(t) in Eq. (13115) with q(t)/i(t) yields

Were getting there. Next we multiply the numerator and denominator of thefirst ratio in Eq. (13116) by i (t), and replace twith our discrete time variable

index nto arrive at our final result of

Figure 1361. Frequency demodulator without arctangent: (a) standardprocess; (b) simplified process.

The implementation of this algorithm, where the derivatives ofi(n) and q(n)

are i(n) and q(n) respectively, is shown in Figure 1361(a). The (n) output

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sequence is used in Eq. (13111) to compute instantaneous frequency.

The Differentiator is an tapped-delay line FIR differentiating filter with an odd

number of taps. Reference [54] reports acceptable results when the dif-

ferentiator is a FIR filter having 1,0,1 as coefficients.

The Delay elements in Figure 1361 are used to time-align i(n) and q(n) with

the outputs of the differentiators such that the delay is (K1)/2 samples whena K-tap differentiator is used. In practice, the Delay can be obtained by

tapping off the center tap of the differentiating filter.

If the i(n)+jq(n) signal is purely FM and hard limitedsuch that i (n)+q (n) =

Constant, the denominator computations in Eq. (13117) need not be

performed. In this case, using the 1,0,1 coefficient differentiators, the FM

demodulator is simplified to that shown in Figure 1361(b) where the Scaling

operation is multiplication by the reciprocal of Constant.

Used with the permission of the publisher, Prentice Hall, this on-going series of articles on

Embedded.com is based on copyrighted material from "Understanding Digital Signal

Processing, Second Edition" by Richard G. Lyons. The book can be purchased on line.

Richard Lyons is a consulting systems engineer and lecturer with Besser Associates. As a

lecturer with Besser and an instructor for the University of California Santa Cruz Extension, Lyons

has delivered digitasl signal processing seminars and training course at technical conferences as

well at companies such as Motorola, Freescale, Lockheed Martin, Texas Instruments, Conexant,

Northrop Grumman, Lucent, Nokia, Qualcomm, Honeywell, National Semiconductor, GeneralDynamics and Infinion.

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http://www.besserassociates.com/http://www.pearson.ch/HigherEducation/PrenticeHallPTR/1449/9780131089891/Understanding-Digital-Signal-Processing.aspx

DSP Tricks_ Frequency Demodulation Algorithms

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