A Multiplicative Cancellation Approach to Multipath Suppression in FM Radio

Abstract

The deceptively simple problem of a single inverted reflection in ordinary frequency modulated (FM) radio is considered. It will be shown that this problem has been overlooked in the literature and causes major breakdown in reception. The problem is known as suppressed-carrier AM-FM (SCAM-FM) and is totally destructive to the received signal. We examine the theory and practical measurements of SCAM and show a solution for reducing its effect.

Appendices

Appendix 1. Simplification of $\psi (t)$

The difference frequency function is given by (16)

$$\begin{aligned} \psi (t)=\beta \sin (\omega _m (t-\pi /\omega _c ))-\beta \sin (\omega _m t)-\pi \end{aligned}$$

(34)

Now the fist sine term in (34) can be written as

$$\begin{aligned} \beta \sin (\omega _m (t-\pi /\omega _c ))=\beta \left[ \sin (\omega _m t)\cos \left( \frac{\pi \omega _m }{\omega _c }\right) -\cos (\omega _m t)\sin \left( \frac{\pi \omega _m }{\omega _c }\right) \right] \end{aligned}$$

which approximates for small angles of $k=\frac{\pi \omega _m }{\omega _c }$ to be

$$\begin{aligned} \beta \sin (\omega _m (t-\pi /\omega _c ))\approx \beta \left[ \sin (\omega _m t)\left( 1-\frac{k^{2}}{2}\right) -k\cos (\omega _m t)\right] \end{aligned}$$

Adding all terms gives in (34)

$$\begin{aligned} \psi (t)\approx \beta \sin (\omega _m t)-\beta k\cos (\omega _m t)-\beta \sin (\omega _m t)-\frac{k^{2}}{2}\beta \sin (\omega _m t)-\pi \end{aligned}$$

or

$$\begin{aligned} \psi (t)\approx -[\pi +\beta k\cos (\omega _m t)]-\frac{k^{2}}{2}\beta \sin (\omega _m t) \end{aligned}$$

Ignoring the second term above as it will be small we get the result

$$\begin{aligned} \psi (t)\approx -(\pi +\beta k\cos (\omega _m t)) \end{aligned}$$

When taking the cosine of $\psi (t)$ we can simplify accordingly

$$\begin{aligned} \cos \psi (t)\approx \cos [-(\pi +\beta k\cos (\omega _m t))]=-\cos [\beta k\cos (\omega _m t))] \end{aligned}$$

and this can be further simplified for small angles to be

$$\begin{aligned} \cos (\psi (t))&= -\cos [\beta k\cos (\omega _m t))]=-\left[ 1-\frac{1}{2}(\beta k)^{2}\cos ^{2}(\omega _m t)\right] \\&= -\left[ 1-\frac{1}{4}(\beta k)^{2}[1+\cos (2\omega _m t)\right] \\&= -1+\frac{1}{4}(\beta k)^{2}[1+\cos (2\omega _m t)] \end{aligned}$$

Hence this function is related to twice the modulating frequency.

Appendix 2. Differentiation of $\frac{d}{dt}\tan ^{-1}\left[ \frac{m\sin (\psi (t))}{1+m\cos (\psi (t))}\right]$

Recall that the differentiation of $\frac{d}{dt}\tan ^{-1}[f(t)]=\frac{1}{1+[f(t)]^{2}}f^{{\prime }}(t)$ where the shorthand notation is used here $f^{{\prime }}(t)=\frac{d}{dt}f(t)$.

To differentiate a function of the form $\frac{u(t)}{v(t)}$ we use the quotient rule

$$\begin{aligned} \frac{d}{dt}\left( \frac{u(t)}{v(t)}\right) =\frac{v(t)du(t)-u(t)dv(t)}{v^{2}(t)} \end{aligned}$$

so that

$$\begin{aligned}&=\frac{d}{dt}\tan ^{-1}\left[ \frac{m\sin (\psi (t))}{1+m\cos (\psi (t))}\right] \\&=\left[ \frac{1}{1+\{\frac{m\sin (\psi (t))}{1+m\cos (\psi (t))}\}^{2}}\right] \frac{d}{dt}\left[ \frac{m\sin (\psi (t))}{1+m\cos (\psi (t))}\right] \\&=\left[ \frac{(1+m\cos (\psi (t))^{2}}{1+2m\cos (\psi (t))+m^{2}}\right] \frac{d}{dt}\left[ \frac{m\sin (\psi (t))}{1+m\cos (\psi (t))}\right] \\&=\left[ \frac{(1+m\cos (\psi (t))^{2}}{1+2m\cos (\psi (t))+m^{2}}\right] \\&\quad \times \left[ \frac{(1+m\cos (\psi (t)))m\cos (\psi (t))-m\sin (\psi (t))(0-m\sin (\psi (t))}{(1+m\cos (\psi (t)))^{2}}\right] \frac{d}{dt}\psi (t) \\&=\left[ \frac{m^{2}+m\cos (\psi (t))}{1+2m\cos (\psi (t))+m^{2}}\right] \frac{d}{dt}\psi (t) \end{aligned}$$

Now

$$\begin{aligned} \psi (t)=\beta \sin (\omega _m (t-\pi /\omega _c ))-\beta \sin (\omega _m t)-\pi \end{aligned}$$

so that its derivative is given by

$$\begin{aligned} \frac{d}{dt}\psi (t)=\beta \omega _m [\cos (\omega _m (t-\pi /\omega _c ))-\cos (\omega _m t)] \end{aligned}$$

hence

$$\begin{aligned}&\frac{d}{dt}\tan ^{-1}\left[ \frac{m\sin (\psi (t))}{1+m\cos (\psi (t))}\right] \\&\quad =\left[ \frac{m^{2}\!+\!m\cos (\psi (t))}{1+2m\cos (\psi (t))+m^{2}}\right] \beta \omega _m [\cos (\omega _m (t\!-\!\pi /\omega _c ))\!-\!\cos (\omega _m t)] \end{aligned}$$