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Frequency Modulation Function
Syntax y = fmmod(m,kf,fc,pc,fs)
Include: include spt\fmmod.oms
See Also ammod , pmmod , quadmod

ARGUMENTS:
   INPUTS:
      m  = MATRIX, any numerical type, coerced to DOUBLE for
           internal processing. Represents sampled version of an
           arbitrary baseband modulation waveform in [Volt].
      kf = SCALAR, any numerical type, coerced to DOUBLE for
           internal processing. Modulator sensitivity in
           [Hz/Volt]. Scales 'm' before modulation.
      fc = SCALAR, any numerical type, coerced to DOUBLE for
           internal processing. Carrier frequency in [Hz].
      pc = SCALAR, any numerical type, coerced to DOUBLE for
           internal processing. Carrier phase in [radians].
      fs = SCALAR, any numerical type, coerced to DOUBLE for
           internal processing. Sampling rate in [Samples/sec].
   RETURN: MATRIX, type DOUBLE, sampled FM waveform. Same 
           dimensions as input 'm'.

Description

Sample a FM (Phase Modulation) modulated carrier with given baseband modulation waveform, carrier frequency and carrier phase in a column-wise fashion.

Returns a sampled version of a FM modulated carrier with the same dimensions as input 'm'. The user specifies the arbitrary baseband modulation waveform 'm' in [Volt], the sampling rate 'fs' in [Samples/sec], the carrier frequency in [Hz], the carrier initial phase angle in [radians], and the modulator sensitivity 'kf' in [Hz/Volt]. All inputs may be any numerical type and are coerced to DOUBLE for local processing.

The general form of the modeled FM waveform is:

     cos(2*PI*fc*t + 2*PI*kf*(integral of m from 0 to t)+ pc,

and this waveform is sampled at a rate of 'fs'. The frequency of the resulting carrier sinusoid is proportional waveform 'm'. 'm' may be a matrix, in which case an identically dimensioned matrix is returned where each column of 'm' has separately been modulated onto the specified carrier. The modulation sensitivity factor 'kf' scales the waveform 'm' before applying it to the carrier and converts the modulation voltage to radians.

It should also be noted that the function places no restrictions on the relation between carrier, modulation, or sampling frequencies. The user may 'under-sample' the waveform by making the maximum frequency represented in 'm' or 'fc' higher than half the sampling rate without error. (This is in fact done deliberately in some sampled data systems).

Example

# Create a FM modulated waveform

fs     = 256d0;          # sampling rate [Samples/second]
N      = 256;            # record length [Samples]
fc     =  32d0;          # carrier frequency [Hz]
pc     =   1d0;          # initial carrier phase [radians]
fm1    =   4d0;          # modulation freq [Hz]
twopit = 2d0*PI*(seq(N)-1d0)/fs; # Create 2*PI*time
m      = cos(twopit*fm1);        # Create modulation waveform
kf     = 1;              # Hz/volt
df     = 2.0*fm1;        # peak freq deviation for freq mod[Hz]
fm     = fmmod(df*m,kf,fc,pc,fs);     # time waveform
spec   = abs(dft(complex(fm))/N)^2d0; # spectrum power
fmspec = spec.row(1,N/2+1); # Form the spectrum for plotting
fmspec = fmspec + {0d0,reverse(spec.row(N/2+1,N/2))};
fmspec = db10(fmspec);                # spectrum mag, dB
t      = timeaxis(1d0/fs, N);         # time axis for plotting
f      = freqaxis(fs/N, N);           # freq axis for plotting

ginit;
format double "f7.1";
gaddtext("FM MODULATOR Function", [.5,.95]);
s=["fs =",ntoa(fs)," Hz"," , fc =",ntoa(int(fc))," Hz"," , fm =",ntoa(int(fm1))," Hz"];
gaddtext(s, [.5,.90]);

vp1 = gaddview(0.05, 0.50, .90, .35 );
gyaxis("linear",-2,2,2,2);
gxaxis("linear",0,N/fs,4,5);
gplot(t,fm);
gxtitle("TIME");
gytitle("VOLTS");
gtitle("fmmod() WAVEFORM");
gygrid("minor");
gxgrid("major");

vp2 = gaddview(.05, .05, .90, .35 );
gyaxis("linear",-60,0,6,2);
gxaxis("linear",0,fs/2,4,4);
gygrid("minor");
gxgrid("major");
gxtitle("FREQ");
gytitle("MAG [dB]");
gtitle("fmmod() SPECTRUM");
gplot(f.row(1,N/2+1),fmspec);

REFERENCE

Haykin, Simon, "Communication Systems.", New York: Wiley, 1994.