FSK Modulation in Python

Question

I am currently writing a script that uses I/Q data to do FSK modulation.

My question is about the FFT plot. I expected to have a peak at 1Hz and 2Hz which are the frequencies that represent binary 0 and 1, however I have peaks at 2 and 4Hz.

I initially suspected that I was using the FFT freq function incorrectly however I set the bin size equal to the number of samples and also defined my sample freq.

Any idea why my FFT frequencies are off?

Thanks in advance

import numpy as np
import matplotlib.pyplot as plt
from scipy.fftpack import fft

binary_in = "10011010" # binary input string
     
sr = 200  # sampling rate
ts = 1.0/sr # sampling interval
time = (np.arange(0,len(binary_in)*1,ts))-ts  # time for plotting entire FSK signal
step = int((len(time))/(len(binary_in)))  # temporal step size for FSK signal relating to each bit
T = np.arange(0,step/100,0.01) # time for calculating sinusoids

f1 = 1  # frequency for binary 0
f2 = 2 # frequency for binary 1

coeff = 2*np.pi

## I and Q defined to have freq of (f1-f2)/2
I = np.cos(coeff*((f1-f2)/2)*T)
Q = np.sin(coeff*((f1-f2)/2)*T)

## In the case of a binary 1 I and Q would be mixed with the LO freq in the following way:
mixer1 = I*np.cos(coeff*((f1+f2)/2)*T)
mixer2 = Q*np.sin(coeff*((f1+f2)/2)*T)

out1 = mixer1+mixer2   #Output is added to give a FSK signal equal to approx. cos(-f2*t)

## In the case of a binary 1 I and Q would be mixed with the LO freq in the following way:
mixer3 = Q*np.cos(coeff*((f1+f2)/2)*T)
mixer4 = I*np.sin(coeff*((f1+f2)/2)*T)

out2 = mixer3+mixer4  #Approx. sin(f1*t)

final_fsk = []  
bin_plot = []  

ones = [1] * step
zeros = [0] * step

j=0 # index for assigning FSK signal to plotted output list


## Assign correct output freq. based on binary input
for i in range(0, len(binary_in)):
    j=j+1
    if binary_in[i] == "1":
        final_fsk[(j*step-(step-1)):(j*step)] = out1
        bin_plot[(j*step-(step-1)):(j*step)] = ones
    else:
        final_fsk[(j*step-(step-1)):(j*step)] = out2
        bin_plot[(j*step-(step-1)):(j*step)] = zeros


fourier = np.fft.fft(final_fsk)
freq=np.fft.fftfreq(len(final_fsk),ts)
# n = np.arange(0, len(fourier))
# period = len(fourier)/sr
# freq = n/period


plt.subplot(311)
plt.title("Binary Input")
plt.plot(np.arange(0, (len(binary_in)),ts), bin_plot)
plt.ylabel('Amplitude')

plt.subplot(312)
plt.title(f"Output FSK for {binary_in}")
plt.plot(time, final_fsk, 'b')
plt.ylabel('Amplitude')
plt.xlabel('Time (s)')

plt.subplot(313)
plt.title("FSK FFT")
plt.plot(freq, np.abs(fourier), 'b')
plt.ylabel('Amplitude')
plt.xlabel('Freq (Hz)')

plt.tight_layout(pad=1)
plt.show()

Answer 1

f1-f2 is a 1 Hz sinusoid, and f1+f2 is 3 Hz sinusoid. Regardless of whether they are cosine or sine, ideal multiplication will only have the sum and difference frequencies in the mixer output: 2 Hz and 4 Hz. The phases could be 0, 90, 180, 270 for the various cases but the frequencies will always be 2 Hz or 4 Hz at the mixer outputs.

Then you add mixer outputs. If the signs were chosen correctly that could enable you to cancel various components, which one assumes was your intent. But addition does not shift any frequencies, so no amount of adding and subtracting can get back the 1 Hz that never left any of the mixers. Your output will always be some combination of 2 Hz and/or 4 Hz, with some smearing due to the binary data not being strictly periodic. You can see in your time domain plot that it's 2 Hz and 4 Hz.

Try f1 = 0.5Hz and f2 = 1Hz if you want 1 Hz and 2 Hz in the output.