PRE- AND DE-EMPHASIS OF AN FM-SIGNAL
Table of Contents
INTRO
The demodulation of an FM-signal is accomplished by differentiating the angle of the received signal which is done here by a delayline demodulator. The transfer-function of a differentiator is proportional to f and therefore the trebles contain an audible amount of noise if the signal has been transmitted linear. Therefore the transmitted signal gets a so called pre-emphasis which boosts the trebles and the received signal has to be processed with the complementary deemphasis to perceive a better SNR of the received signal.
TIME DOMAIN
The curve of the analog transfer function can be seen in the bode-diagram described as analog pre-emphasis. It can be synthesized of a parallel circuit of a P-element 
and a D-element (differentiator). The transfer function of a D-element in general ist 
. If a crossing point at the frequecy of
is desired, the transfer function has to be 
.
and a D-element (differentiator). The transfer function of a D-element in general ist . If a crossing point at the frequecy of is desired, the transfer function has to be . Instead of a cutoff-frequency the pre-emphasis is specified as a time constant with 
in Europe. In America 
which means that the frequency boosts starts at lower frequencies. The corresponding cutoff frequency 
in Europe. In America which means that the frequency boosts starts at lower frequencies. The corresponding cutoff frequency In the bode-diagram 
results in a horizontal line at 0 dB and 
is a line with a slope of 20 dB/Decade with an appropriate 0 dB crossing. Therefore the parallel circuit can be written as 
. At the cutoff-frequency the magnitude of the signal is 
results in a horizontal line at 0 dB and is a line with a slope of 20 dB/Decade with an appropriate 0 dB crossing. Therefore the parallel circuit can be written as . At the cutoff-frequency the magnitude of the signal is The digital implementation is quite straight forward by replacing the analog D-element with a digital differentiator. The difference equation of such an element is 
.
.Again the slope has to be adjusted by the factor 
.
. 
. The difference between an analog and a digital pre-emphasis can be seen in the bode-diagram.
FREQUENCY DOMAIN
The transfer function of the difference equation of the pre-emphasis can be obtained by a z-transformation.
The de-emphasis has the complementary transfer-function
The frequency response can be obtained by replacing z by 
which leads to the Fourier transform of the signal. The bodeplot command in Matlab is able to draw bode-diagrams of analog and digital transfer functions, therefore this step will be skipped.
which leads to the Fourier transform of the signal. The bodeplot command in Matlab is able to draw bode-diagrams of analog and digital transfer functions, therefore this step will be skipped.IMPLEMENTATION
%% Design of digital pre- and de-emphasis filter for an FM-radio
% Declarations
fs=240e3;
fc=1/(2*pi*50e-6); %tau=50µs in Europe
fmax=15e3;
k=fs/(2*pi*fc);
% transfer functions of analog and digital filters
pre_analog=tf([1/(2*pi*fc) 1], 1,'variable','s');
pre_digital=tf([1+k -k],[1 0],1/fs,'variable','z^-1')
de_digital=tf([1 0],[1+k -k],1/fs,'variable','z^-1')
pre_n_de=pre_digital*de_digital;
% Bode-Diagram of all filters in comparison
h=bodeplot(pre_analog,pre_digital,de_digital,pre_n_de);
grid on
setoptions(h,'FreqUnits','Hz')
legend('analog pre-emphasis','digital pre-emphasis','digital de-emphasis','digital pre and de');
