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 
117 
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. 
119 
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  .