لخّصلي

Frequency Modulation
Frequency modulation (FM) was introduced by E.A. Armstrong in the
1930’s as an alternative to the AM commonly in use at the time for
broadcasting. The advantage to frequency modulation is that, for a given
transmitted power, the signal-to-noise ratio is much higher at the receiver
output than it is for AM. The digital version of FM, frequency-shift
keying, has been in use since an even earlier date.
In FM, the frequency of the carrier is modulated to follow the amplitude
of the message signal. To be more specific, if  
mt
with peak value p
m
is a message signal
, then the instantaneous frequency  
f t
is given by
 
f t f kmt
 
 ,
c f
of the carrier
(17)
where c
f is the carrier frequency and f
k is a proportionality constant
called the “frequency sensitivity.” The term
k m t
f
 
is called the
“frequency deviation” of the instantaneous frequency from the carrier
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118

frequency, and the peak frequency deviation f p
f km   is an important
FM system parameter. Given the instantaneous frequency, we can find
the total instantaneous angle   t  of the carrier by integrating the
instantaneous frequency. That is,

   
   
0
0
2
2 2 0.
t
t
c f
t f d
f t k m d
   
    

  


(18)
Since the initial angle   0  is of no consequence, we can simplify the
equations by taking   0 0   . The transmitted FM signal is then given by

   
 
 
0
0
cos
cos 2 2
cos 2 2 .
c
t
c c f
t
c c p
g t A t
A f t k m d
A f t f m m d

   
   
    
       
            


(19)
Figure 1 shows a 2 kHz carrier frequency modulated by a 200 Hz
sinusoidal message.
Figure 1. Frequency Modulated Signal
To create an FM signal using the USRP, the message signal is
normalized to a peak value of one, multiplied by 2 f

give  
2 t
p

  
f m m d
 
.
and integrated to
Next, the complex-valued signal  
0
  
formed, where

t
j

   
0

f m m d
g t Ae   
 
  
c
The complex-valued signal  
g t
2 .
p
g t
is
(20)
is sent to the Write Tx Data vi, and the
USRP produces the FM signal.
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120

The only tricky step in generating an FM signal is integrating the
message. In discrete time we have
    0 0
, n t
k
x d x nT T  

  (21)
where T is the reciprocal of the IQ sample rate. If we write
   
0
, n
k
yn x nT T

 (22)
then
       
 
1
0 0
1
.
n n
k k
y n y n x nT T x nT T
x nT T

 
   

  (23)
Equation (23) is the difference equation of an IIR filter. This filter can
be implemented using the IIR Filter vi found in the Signal
ProcessingFiltersAdvanced IIR Filtering palette. Use a “forward
coefficients” array of   T and a “reverse coefficients” array of   1 1  .