FM调制解调

FM调制是是恒包络调制，基本没有峰均比，PA利用率高，相对于幅度调制有更好的抗干扰性能，缺点是带宽利用率低。

FM调制信号表示如下，其实质是差分频率调制，即该sample的频率是前一个sample的频率加上调制信号变化量，具体表现为每个sample相位的变化。

假设| x(t) | ≤ 1，则fd是相对于中心频点的最大频偏。当x(t)恒等于1时，等效为上变频fd。

FM调制通常会拓展频谱，这是因为频率调制的本质是搬频，占用带宽为最大的搬频距离加上基带带宽，即R+2*fd，其中R为基带信号带宽（符号速率），调制因子dm=fd/fmax，fmax=R/2，| x(t) | ≤ 1

FM解调使用差分解调，提取相位变化量。

根据FM调制原理，t时刻调制的相位为

2*pi*fd* x(t-1)/fs+2*pi*fd*x(t)/fs

其中dt = 1/fs，相位增量为2*pi*fd*x(t)/fs，需要满足在[-pi pi]的范围内，即x(t)*fd/fs的绝对值应小于0.5，否则会造成相位溢出失真。

在保证相位无失真的情况下调整x(t)的增益可直接调整调制信号带宽。

FM实现需要框图如下，解调时的fs可以不等于调制时的fs，具体通过延时N来调整，不能让相位溢出

频谱仿真

%% Plot Spectrum of FM Modulated Baseband Signal

% Apply FM baseband modulation to BPSK source and plot its

% spectrum.

clearclose all

% Set the example parameters.

fs = 20e3              % Sample rate (Hz)

ts = 1/fs              % Sample period (s)

Rs = 1e3              % Baseband symbol rate

dm = 1                % FM modulation factor

fd = Rs/2*dm          % Frequency deviation (Hz)

Nup = fs/Rs            % Up-sampling times

%%

% Create a BPSK symbol source having a duration of 1s.

%%

Nsymb = 1*Rs

symbBpsk = (randi([0 1],Nsymb, 1)-0.5)*2

%%

% Upsampleing BPSK symbol to fs and shape it by Raised cosine FIR pulse-shaping filter

%%

rrcFilter = rcosdesign(0.25,20,Nup,'norm')

dataRrcIn = upsample(symbBpsk,Nup)

dataRrcTemp = conv(dataRrcIn,rrcFilter)

NtailRrc = floor(length(rrcFilter)/2)

dataRrcOut  = dataRrcTemp(NtailRrc+1:end-NtailRrc)

% Scale power, normalise the power of optimum sampling point, the scale

% of FM input signal will directly determine FM bandwidth

xOptSamp = dataRrcOut(1:Nup:end)

plot(xOptSamp,'*')

x = dataRrcOut/mean(abs(xOptSamp))

plot(x,'*')

%%

% Create an FM modulator System object and modulate the input signal.

%%

MOD1 = comm.FMModulator('SampleRate',fs,'FrequencyDeviation',fd)

y = step(MOD1,x)

%%

% Create another modulator object, |MOD2|, whose frequency deviation is

% two times larger and apply FM modulation.

%%

MOD2 = comm.FMModulator('SampleRate',fs,'FrequencyDeviation',2*fd)

z = step(MOD2,x)

%%

% Plot the spectra of the two modulated signals. The larger frequency deviation

% associated with channel 2 results in a noise level that is 10 dB higher.

%%

SA = dsp.SpectrumAnalyzer('SampleRate',fs,'ShowLegend',true)

step(SA,[x y z])

%%

调整信号大小仿真

% FM link

clearclose all

fb = 1e3 

dm = 1

fd = Rs/2*dm          % Frequency deviation (Hz)

fs = 8e3

Nup = fs/fb

data = (randi([0 1],1000, 1)-0.5)*2  % BPSK

figureplot(data,'-*')

% 成型滤波器

rrcFilter = rcosdesign(0.25,20,Nup,'norm')

dataRrcIn = upsample(data,Nup)

dataRrcTemp = conv(dataRrcIn,rrcFilter)

NtailRrc = floor(length(rrcFilter)/2)

dataRrcOut  = dataRrcTemp(NtailRrc+1:end-NtailRrc)

dataIn = dataRrcOut/mean(abs(dataRrcOut))      % Scale power

% 调制相位需要控制在[-pi pi],否则会造成相位失真，即 dataIn*fd/fs的绝对值不能超过0.5

dataIn = dataIn*fs/fd*0.5   

maxDelta = max(abs(dataIn*fd/fs))

FMMethod = comm.FMModulator('SampleRate',fs,'FrequencyDeviation',fd)

modSymb = step(FMMethod,dataIn)

FMDeMethod = comm.FMDemodulator('SampleRate',fs,'FrequencyDeviation',fd)

demodSymb = step(FMDeMethod,modSymb)

isequal(fi(dataIn,1,13,10),fi(demodSymb,1,13,10))

return

% my function

addTemp = 0

for i = 1:length(dataIn)

    addTemp = dataIn(i)+addTemp

    modPhase(i,1) = 2*pi*fd*addTemp*(1/fs)

end

modSymbOut = exp(1j*modPhase)

dataDemodIn = modSymbOut

% demod

dataDemodOut(1,1) = angle(dataDemodIn(1))/(2*pi*fd/fs)

Ndelay = 1

for i = 1:length(dataDemodIn)-Ndelay

    deltaPhase = dataDemodIn(i+Ndelay)*conj(dataDemodIn(i))

    dataDemodOut(i+1,1) = angle(deltaPhase)/(2*pi*fd/fs)

end

isequal(fi(dataIn,1,13,10),fi(dataDemodOut,1,13,10))

先得信号的相位，unwrap 校正相位角，使相位在整体上连续，diff对该相位角求导至于线性计算部分跟你的调制公式有关系的，不一定。

u=cos(2*pi*fc*t+2*pi*kf*int_m)

int_m是0—t区间内对信号函数的积分。

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