DSP Tricks: Frequency demodulation algorithms

Richard Lyons

January 10, 2011

Richard LyonsJanuary 10, 2011

An often used technique for measurement of the instantaneous fre­quency of a complex sinusoidal signal is to compute the derivative of the signal’s instantaneous θ(n) phase as shown in Figure 13–60 below.


Figure 13–60  Frequency demodulator using an arctangent function.

This is the traditional discrete signal FM demodulation method, and it works fine. The demodulator’s instantaneous output frequency is Eq13-111 below:


where fsis the sample rate in Hz.

Computing instantaneous phase θ(n) requires an arctangent operation, which is difficult to implement accurately without considerable computa­tional resources. Here’s a scheme for computing ∆θ(n) for use in Eq. (13–111) without the intermediate θ(n) phase computation (and its pesky arctan-gent). We derive the ∆θ(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,
θ(t) = tan–1[q(t)/i(t)] = instantaneous phase,
∆θ(t) = time derivative of θ(t). (13-112) 

First, we let r(t)= q(t)/i(t) be the signal for which we’re trying to compute the derivative of its arctangent. The time derivative of tan–1[r(t)], a calculus iden­tity, is

Because d[r(t)]/dt = d[q(t)/i(t)]/dt, we use the calculus identity for the deriv­ative of a ratio to write

Plugging Eq. (13–114)’s result into Eq. (13–113), we have

Replacing r(t) in Eq. (13–115) with q(t)/i(t) yields

We’re getting there. Next we multiply the numerator and denominator of the first ratio in Eq. (13–116) by i2(t), and replace twith our discrete time variable index nto arrive at our final result of


Figure 13–61. Frequency demodulator without arctangent: (a) standard process; (b) simplified process.

The implementation of this algorithm, where the derivatives of i(n) and q(n) are i’(n) and q’(n) respectively, is shown in Figure 13–61(a). The ∆φ(n) out­put sequence is used in Eq. (13–111) 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 dif­ferentiator is a FIR filter having 1,0,–1 as coefficients.

The Delay elements in Figure 13–61 are used to time-align i(n) and q(n) with the outputs of the dif­ferentiators such that the delay is (K–1)/2 samples when a 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 limited such that i2(n)+q2(n) = Constant, the denominator computations in Eq. (13–117) need not be performed. In this case, using the 1,0,–1 coefficient differentiators, the FM demodulator is simplified to that shown in Figure 13–61(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, General Dynamics and Infinion.

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