Implementing DFT, inverse function not working properly

Question

I've implemented DFT and the inverse DFT Function according to the following formulas:

The DFT function works, but when testing the inverse on the output I don't get the original series.

import numpy as np
import random

exp = np.exp
pi = np.pi

def mydft(X):
    n = len(X)
    out = []
    for k in range(n):
        temp = 0
        for i in range(n):
            temp += X[i] * exp(-2j*pi*k*i/n)
        out.append(temp)
    return np.array(out)

def myidft(X):
    n = len(X)
    out = []
    for k in range(n):
        temp = 0
        for i in range(n):
            temp += X[i] * exp(2j*pi*k*i/n)
        out.append(temp)
    return (1/n) * np.array(out)

Testing

orig = np.random.random(100)
trans = mydft(orig)

inv = myidft(trans)
print(np.allclose(inv, trans))
>>> False

print(np.allclose(trans, np.fft.fft(orig)))
>>> True

Since the original Function works and the modifications for the inverse are quite simple, I have no clue what went wrong!?

Answer 1

Shouldn't your test be

print(np.allclose(inv, orig))

since

orig = myidft(mydft(orig))

because when I plot your DFT

and invDFT (of DFT of original signal)

compared to the numpy FFT algorithm the results are exactly the same. Your implementation seems right. Your test is wrong.