# Figure of merit of FM

The block diagram of FM Receiver in the presence of noise is as follows

The incoming signal at the front end of the receiver is an FM signal $S_{FM}(t)&space;=&space;A_{c}&space;cos(2\pi&space;f_{c}t+2\pi&space;k_{f}\int&space;m(t&space;)dt&space;)--------Equation(1)$ got interfered by Additive noise $n(t)$, since the FM signal has a transmission band width $B_{T}$,the Band Pass filter characteristics are also considered over the band of interest i.e from$\frac{-B_{T}}{2}$ to $\frac{B_{T}}{2}$.

The output of Band Pass Filter is $x(t)&space;=&space;S_{FM}(t)+n_{o}(t)------------------Equation(2)$ is passed through a Discriminator for simplicity simple slope detector (discriminator followed by envelope detector) is used, the output of discriminator is $v(t)$ this signal is considered over a band of $(-W,W)$ by using a LPF .

The input noise to the BPF is n(t),  the  resultant output noise is band pass noise $n_{o}(t)$

$n_{o}(t)&space;=&space;n_{I}(t)cos&space;\omega&space;_{c}t-n_{Q}(t)sin&space;\omega&space;_{c}t-----------Equation(3)$

phasor representation of Band pass noise is $n_{o}(t)=r(t)cos&space;(\omega&space;_{c}t&space;+\Psi&space;(t))$ where $r(t)=\sqrt{n_{I}^{2}(t)+n_{Q}^{2}(t)}$ and $\Psi&space;(t)&space;=&space;\tan&space;^{-1}(\frac{n_{Q}(t)}{n_{I}(t)})$.

$n_{I}(t),n_{Q}(t)$ are orthogonal, independent and are Gaussian.

$r(t)$– follows a Rayleigh’s distribution and $\Psi&space;(t)$ is uniformly distributed over $(0,2\pi&space;)$$r(t)&space;and&space;\Psi&space;(t)$ are separate random processes.

substituting Equations (1), (3) in (2)

$x(t)&space;=&space;S_{FM}(t)+n_{o}(t)$

$x(t)=&space;A_{c}&space;cos(2\pi&space;f_{c}t+2\pi&space;k_{f}\int&space;m(t&space;)dt&space;)+n_{I}(t)cos&space;\omega&space;_{c}t-n_{Q}(t)sin&space;\omega&space;_{c}t$

$x(t)=&space;A_{c}&space;cos(2\pi&space;f_{c}t+2\pi&space;k_{f}\int&space;m(t&space;)dt&space;)+r(t)cos&space;(\omega&space;_{c}t&space;+\Psi&space;(t))$

$x(t)=&space;A_{c}&space;cos(2\pi&space;f_{c}t+\Phi&space;(t)&space;)+r(t)cos&space;(\omega&space;_{c}t&space;+\Psi&space;(t))-----Equation(4)$ where $\Phi&space;(t)=2\pi&space;k_{f}\int&space;m(t&space;)dt$.

now the analysis is being done from it’s phasor diagram/Noise triangle as follows

$x(t)$ is the resultant of two phasors $A_{c}&space;cos(2\pi&space;f_{c}t+\Phi&space;(t)&space;)$ and $r(t)cos&space;(\omega&space;_{c}t&space;+\Psi&space;(t))$.

$\theta&space;(t)-\Phi&space;(t)&space;=&space;\tan&space;^{-1}(\frac{r(t)\sin&space;(\Psi&space;(t)-\Phi&space;(t))}{A_{c}+r(t)\cos&space;(\Psi&space;(t)-\Phi&space;(t))})$

$\theta&space;(t)-\Phi&space;(t)&space;=&space;\tan&space;^{-1}(\frac{r(t)\sin&space;(\Psi&space;(t)-\Phi&space;(t))}{A_{c}})$ since $r(t)<&space;

$\theta&space;(t)=\Phi&space;(t)&space;+\tan&space;^{-1}(\frac{r(t)\sin&space;(\Psi&space;(t)-\Phi&space;(t))}{A_{c}})$

$\theta&space;(t)=\Phi&space;(t)&space;+\frac{r(t)\sin&space;(\Psi&space;(t)-\Phi&space;(t))}{A_{c}}$ because $\frac{r(t)}{A_{c}}<&space;<&space;1\Rightarrow&space;\tan&space;^{-1}\theta&space;=\theta$.

$\theta&space;(t)$ is the phase of the resultant signal $x(t)$ and when this signal is given to a discriminator results  an output$v(t)$.

i.e, $v(t)=\frac{1}{2\pi&space;}\frac{d\theta&space;(t)}{dt}$

i.e, $v(t)=&space;\frac{1}{2\pi&space;}\frac{d}{dt}(\Phi&space;(t)&space;+\frac{r(t)\sin&space;(\Psi&space;(t)-\Phi&space;(t))}{A_{c}})---------Equation(5)$

As $\Phi&space;(t)&space;=2\pi&space;k_{f}\int&space;m(t)dt$

$\frac{d\Phi&space;(t)}{dt}=2\pi&space;k_{f}m(t)$

the second term in the Equation $n_{d}(t)=\frac{1}{2\pi&space;}\frac{d}{dt}(\frac{r(t)\sin&space;(\Psi&space;(t)-\Phi&space;(t))}{A_{c}})$ where $n_{d}(t)$ – denotes noise after demodulation.

this can be approximated to $n_{d}(t)=\frac{1}{2\pi&space;A_{c}&space;}\frac{d}{dt}(r(t)\sin&space;\Psi&space;(t))-------Equation(6)$, which is a valid approximation. In this approximation $r(t)\sin&space;\Psi&space;(t)$ is Quadrature-phase noise with power spectral density $S_{NQ}(f)$ over $(\frac{-B_{T}}{2},\frac{B_{T}}{2})$

the power spectral density of $n_{d}(t)$ will be obtained from Equation (6) using Fourier transform property $\frac{d}{dt}\leftrightarrow&space;j2\pi&space;f$

$S_{Nd}(f)=\frac{1}{(2\pi&space;A_{c})^{2}}(2\pi&space;f)^{2}S_{NQ}(f)$

$S_{Nd}(f)=(\frac{f}{A_{c}})^{2}S_{NQ}(f)$  , $\left&space;|&space;f&space;\right&space;|\leq&space;\frac{B_{T}}{2}$

$S_{Nd}(f)=0$    elsewhere.

the power spectral density functions are drawn in the following figure

$\therefore&space;v(t)&space;=&space;k_{f}m(t)+n_{d}(t)-------Equation(7))$ , from Carson’s rule $\frac{B_{T}}{2}\geq&space;W$

the band width of v(t) has been restricted by passing it through a LPF.

Now, $S_{Nd}(f)=(\frac{f}{A_{c}})^{2}S_{NQ}(f),\left&space;|&space;f&space;\right&space;|\leq&space;W$

$S_{Nd}(f)=0&space;elsewhere$.

To calculate Figure of Merit $FOM&space;=&space;\frac{(SNR)_{output}}{(SNR)_{input}}$

Calculation of $(SNR)_{output}$:-

output Noise power $P_{no}&space;=&space;\int_{-W}^{W}(\frac{f}{A_{c}})^{2}&space;N_{o}df$

$P_{no}&space;=&space;\frac{N_{o}}{A_{c}^{2}}&space;\left&space;(&space;\frac{f^{3}}{3}&space;\right&space;)^{W}_{-W}$

$P_{no}&space;=&space;\frac{N_{o}}{A_{c}^{2}}&space;\left&space;(&space;\frac{2W^{3}}{3}&space;\right&space;)------Equation(I)$

The output signal power is calculated from $k_{f}m(t)$ tha is  $P_{so}&space;=&space;k_{f}^{2}P--------Equation(II)$

$(SNR)_{output}&space;=&space;\frac{P_{so}}{P_{no}}$

From Equations(I) and (II)

$(SNR)_{output}&space;=\frac{\frac{N_{o}}{A_{c}^{2}}&space;\left&space;(&space;\frac{2W^{3}}{3}&space;\right&space;)}{k_{f}^{2}P}$

$(SNR)_{output}&space;=\frac{3}{2}\frac{k_{f}^{2}PA_{c}^{2}}{N_{o}W^{3}}-------Equation(8)$

Calculation of $(SNR)_{input}$:-

$(SNR)_{input}&space;=&space;\frac{P_{si}}{P_{ni}}$

input signal power $P_{si}=&space;\frac{A_{c}^{2}}{2}---------------Equation(III)$

noise signal power  $P_{ni}=N_{o}W--------------Equation(IV)$

from Equations (III) and (IV)

$(SNR)_{input}&space;=&space;\frac{A_{c}^{2}}{2WN_{o}}-------------------Equation(9)$

Now the Figure of Merit of FM is $FOM&space;=&space;\frac{(SNR)_{output}}{(SNR)_{input}}$

$FOM&space;=&space;\frac{\frac{3}{2}\frac{k_{f}^{2}PA_{c}^{2}}{N_{o}W^{3}}}{\frac{A_{c}^{2}}{2WN_{o}}}$

$FOM_{FM}&space;=&space;\frac{3k_{f}^{2}P}{W^{2}}--------Equation(10)$

to match this with AM tone(single-tone) modulation is used i.e, $m(t)&space;=&space;cos&space;\omega&space;_{m}t$ then the signal power $P&space;=&space;\frac{1}{2}$  and $W&space;=&space;f_{m}$

$FOM_{FM}&space;=&space;\frac{3k_{f}^{2}}{f_{m}^{2}}\frac{1}{2}$

$FOM_{FM}&space;=&space;\frac{3}{2}&space;(\frac{k_{f}}{f_{m}})^{2}$

$FOM_{FM}&space;=&space;\frac{3}{2}\beta&space;^{2}$

since for tone(single-tone) modulation $\beta&space;=&space;\frac{k_{f}}{f_{m}}$.

when you compare single-tone FM with AM $FOM_{FM&space;(single-tone)}&space;=FOM_{AM(single-tone)}$

$\frac{3}{2}\beta&space;^{2}&space;>&space;\frac{1}{3}$

$\beta&space;>&space;\frac{\sqrt{2}}{3}$

$\beta&space;>&space;0.471$.

the modulation index $\beta&space;>0.471.$ will be beneficial in terms of noise cancellation, this is one of the reasons why we prefer WBFM over NBFM.

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