Scipy FFT and Numpy FFT disagree on pulse train spectrum?

Question

I am doing an FFT on a series of pulses. The series is one pulse of amplitude 1 every 7 days over a total of 367 days. The code below is what I run:

import pandas as pd
from scipy.fft import fft, fftfreq, fftshift, ifft
from scipy.signal import blackman
from matplotlib import pyplot as plt
import random

## Signal 
num_samples = 367
# time in days
t = np.arange(int(num_samples))
# Amplitude and position of pulse. Amplitude here is 0 or 1 but can generate random values
# Position here is every 7th day
signal = [random.randint(1,1) if (i%7 == 0) else 0 for i, x in enumerate(t)]#np.sin(2*np.pi*5*t/N)#[random.randint(1,1) if (i%7 == 0) else 0 for i, x in enumerate(t)]#

# FFT and IFFT using Numpy

sr = 367
X = np.fft.fft(signal)
n = np.arange(num_samples)
T = num_samples/sr
freq = n/T 

plt.figure(figsize = (12, 6))
plt.subplot(121)
plt.title('FFT using Numpy')
plt.stem(freq, np.abs(X), 'b', markerfmt=" ", basefmt="-b")
plt.xlabel('Freq (Hz)')
plt.ylabel('FFT Amplitude |X(freq)|')

plt.subplot(122)
plt.title('IFFT using Numpy')
plt.plot(t, np.fft.ifft(X), 'r')
plt.xlabel('Time (s)')
plt.ylabel('Amplitude')
plt.tight_layout()
plt.show()

# FFT and IFFT using Scipy

sp = fft(signal)
freq = fftfreq(t.shape[-1])

plt.figure(figsize = (12, 6))
plt.subplot(121)
plt.title('FFT using Scipy')
plt.stem(freq, np.abs(sp), 'b', markerfmt=" ", basefmt="-b")
plt.xlabel('Freq (Hz)')
plt.ylabel('FFT Amplitude |sp(freq)|')

plt.subplot(122)
plt.title('IFFT using Scipy')
plt.plot(t, ifft(sp), 'r')
plt.xlabel('Time (s)')
plt.ylabel('Amplitude')
plt.tight_layout()
plt.show()

This results in the following plots:

So I'm confused as to what is happening:

• I did not expect any peaks but rather a 'comb' b/c it's fft on pulse train
• if these are correct peaks then I would expect the closest to 0 frequency (7-day period) to taller than the ones to its right
• it looks like there is a scaling/shifting issue

Any guidance would be appreciated.

Answer 1

If you change your line:

plt.stem(freq, np.abs(X), 'b', markerfmt=" ", basefmt="-b")

to

plt.stem(freq, fftshift(np.abs(X)), 'b', markerfmt=" ", basefmt="-b")

the graphs will be the same for both.

The reason for the "extra" spikes is that the length of the signal (367) is not divisible by 7. If that is increased to 371, then I get the following graphs is is probably closer to what you're expecting.

---

Code update to see interpolation effect

import numpy as np
import pandas as pd
from scipy.fft import fft, fftfreq, fftshift, ifft
from scipy.signal import blackman
from matplotlib import pyplot as plt
import random

## Signal 
num_samples = 371
# time in days
t = np.arange(int(num_samples))
t3 = np.arange(int(num_samples)*3)
# Amplitude and position of pulse. Amplitude here is 0 or 1 but can generate random values
# Position here is every 7th day
signal = [random.randint(1,1) if (i%7 == 0) else 0 for i, x in enumerate(t)]#np.sin(2*np.pi*5*t/N)#[random.randint(1,1) if (i%7 == 0) else 0 for i, x in enumerate(t)]#

# FFT and IFFT using Numpy

sr = 367
X = np.fft.fft(signal, num_samples*3)
n = np.arange(len(X))
T = num_samples/sr
freq = n/T 

plt.figure(figsize = (12, 6))
plt.subplot(121)
plt.title('FFT using Numpy')
plt.stem(freq, fftshift(np.abs(X)), 'b', markerfmt=" ", basefmt="-b")
plt.xlabel('Freq (Hz)')
plt.ylabel('FFT Amplitude |X(freq)|')

plt.subplot(122)
plt.title('IFFT using Numpy')
plt.plot(t, np.fft.ifft(X)[0:num_samples], 'r')
plt.xlabel('Time (s)')
plt.ylabel('Amplitude')
plt.tight_layout()
plt.show()

# FFT and IFFT using Scipy

sp = fft(signal, num_samples*3)
freq = fftfreq(t.shape[-1]*3)

plt.figure(figsize = (12, 6))
plt.subplot(121)
plt.title('FFT using Scipy')
plt.stem(freq, np.abs(sp), 'b', markerfmt=" ", basefmt="-b")
plt.xlabel('Freq (Hz)')
plt.ylabel('FFT Amplitude |sp(freq)|')

plt.subplot(122)
plt.title('IFFT using Scipy')
plt.plot(t, ifft(sp)[0:num_samples], 'r')
plt.xlabel('Time (s)')
plt.ylabel('Amplitude')
plt.tight_layout()
plt.show()