How to convert from frequency domain to time domain in python

Question

I know this is basic for signal processing, but, I am not sure what is wrong about my approach. I have a signal that behaves as damped sine signal with a sampling frequency of 5076Hz and 15,000 number of samples. I found from the following website how to convert a signal from a time domain to frequency domain and managed to get the FFT and frequency values. The code can be found below the link:

Machine Learning with Signal Processing Techniques

def get_fft_values(y_values, T_s, N, f_s):
    f_values = np.linspace(0.0, 1.0/(2.0*T), N//2)
    fft_values_ = np.fft.rfft(y_values)
    fft_values = 2.0/N * np.abs(fft_values_[0:N//2])
    return f_values, fft_values

I managed to get the frequency and FFT values. However, I need to implement filters to remove some noise from the signal, so, I created the following functions to implement the filter part:

def butter_bandpass(lowcut, highcut, fs, order):
    nyq = 0.5 * fs
    low = lowcut / nyq
    high = highcut / nyq
    b, a = butter(order, [low, high], btype='bandpass', output='ba')
    return b, a

def butter_bandpass_filter(data, lowcut, highcut, fs, order):
    b, a = butter_bandpass(lowcut, highcut, fs, order=order)
    y = filtfilt(b=b, a=a, x=data)
    # y = lfilter(b=b, a=a, x=data)
    return y

I know that I would need to implement the following steps:

• Convert to the frequency domain
• apply a bandpass filter to get rid of frequencies you don't care about
• convert back to the time domain by inverse Fourier transform

So, I created the following inverse transform function, but, I can't get the filtered signal back and the amplitudes don't almost match the original signal. (For my case, I need to resample)

def get_ifft_values(fft_values, T, N, f_s):
    # Time axis:
    N = 9903
    S_T = 1 / S_F
    t_n = S_T * N  # seconds of sampling
    # Obtaining data in order to plot the graph:
    x_time = np.linspace(0, t_n, N)
    ifft_val = np.fft.irfft(fft_values, n=N)
    y_s, x_time = scipy.signal.resample(x=ifft_val, num=N, t=x_time)
    return x_time, y_s

What am I doing wrong here?

Edit 1:

Based on the answer from @Han-Kwang Nienhuys. I edited the above code and applied it to the approach below:

##### Converting the signal into fft:
f_val, fft_val = get_fft_values(y_values=y, T=S_T, N=N, f_s=S_F)
# Applying bandpass filter:
fft_filt_val = butter_bandpass_filter(data=fft_val, lowcut=50, highcut=600, fs=S_F, order=2)
# Applying the inverse transform of the frequency domain:
x_time, y = get_ifft_values(fft_values=fft_filt_val, T=S_T, N=N, f_s=S_F)

Here are the results from the signal:

• FFT of the original signal:

• Filtered FFT of the original signal:

• Converted Signal from Filtered FFT:

• Without Applying the bandpass filter:

Answer 1

There are several issues:

• You are using np.fft.fft , which is a complex-valued discrete Fourier transform, containing frequencies up to twice the Nyqvist frequency. The frequencies above the Nyqvist frequency can be interpreted equivalently as negative frequencies. You are indeed using a frequency slice [:N//2] , but if you want the inverse transform to work, you also need to deal with the other half of the spectrum.
• Don't take the absolute values of the FFT data. The filter must operate on the complex-valued coefficients.
• If you use scipy.signal.filtfilt : that function operates on time-domain data, not on frequency-domain data.

For real-valued input data, it's much easier to use a real-valued FFT, which will behave more like you expect:

n = len(y)
yf = np.fft.rfft(y)
fstep = f_sampling / n
freqs = np.arange(len(yf)) * fstep

To transform back, use np.fft.irfft .