Inverse Fourier transform in Python not yielding the original signal

Question

I am trying to do a Fourier Transform and Inverse Fourier Transform. The problem is that I am not getting the original signal after inverse transform.

# set signal and transform constants
samples = 10000
t = np.arange(samples)
cf = 1000 # carrier frequency Hz
mf = 60 # modulation frequency Hz
# signal is a classic AM waveform
signal = lambda t: np.sin(2 * np.pi * cf * t / samples) \
* (1+(np.sin(2 * np.pi * mf * t / samples))) 


fi = np.arange(5000)
fs = np.fft.fft(signal(t))[:5000]
sp2.plot(fi ,abs(fs) * 2 / samples, color = "#fb8072")



samples = 10000
t = np.arange(samples)
# declare an all-zeros frequency spectrum
s = np.zeros((samples,), dtype=complex)
# set the spectral lines
s[940] = 0.5
s[1000] = 1.0
s[1060] = 0.5


fs = np.fft.ifft(s)[:512]
sp2.plot(fi, fs * samples ,color='#fdb462')
plt.grid(color='grey', linestyle='-.', linewidth=0.5)
plt.show()

Why is the yellow time-domain signal not as the same as the original blue signal?

Answer 1

Your code:

samples = 10000
t = np.arange(samples)
# declare an all-zeros frequency spectrum
s = np.zeros((samples,), dtype=complex)
# set the spectral lines
s[940] = 0.5
s[1000] = 1.0
s[1060] = 0.5


fs = np.fft.ifft(s)[:512]
sp2.plot(fi, fs * samples ,color='#fdb462')
plt.grid(color='grey', linestyle='-.', linewidth=0.5)
plt.show()

seems to be making the one-sided spectrum that your first signal creates.

For the inverse to work, your Fourier domain signal needs to have both positive and negative frequency peaks.

You also haven't clarified Dan's question: why are you doing that, rather than just

s = np.fft.ifft(fs)

given the fs is the frequency domain representation of your original signal.

That part doesn't make sense to me.