Deriving the Inverse Fast Fourier Transform (IFFT)

The Inverse Fast Fourier Transform (IFFT) is a fundamental algorithm in digital signal processing and data analysis. It is derived from the principles of both the Discrete Fourier Transform (DFT) and the Fast Fourier Transform (FFT) algorithms. This article provides a detailed explanation of the derivation of the IFFT, including its relationship with the DFT and FFT.

1. Discrete Fourier Transform (DFT)

The Discrete Fourier Transform (DFT) converts a sequence of complex numbers ( x[n] ) into another sequence of complex numbers ( X[k] ). The DFT is defined as:

[ X[k] sum_{n0}^{N-1} x[n] expleft(-i frac{2pi}{N} knright) text{ for } k 0, 1, ldots, N-1 ]

In this formula:

x[n] represents the input sequence of complex numbers in the time domain. N is the number of points in the transform. i is the imaginary unit. exp(-i frac{2pi}{N} kn) represents the complex exponential basis functions.

2. Inverse Discrete Fourier Transform (IDFT)

The Inverse Discrete Fourier Transform (IDFT) is used to convert the frequency domain representation ( X[k] ) back to the time domain representation ( x[n] ). The IDFT is defined as:

[ x[n] frac{1}{N} sum_{k0}^{N-1} X[k] expleft(i frac{2pi}{N} knright) text{ for } n 0, 1, ldots, N-1 ]

3. Derivation of IFFT from DFT

The IFFT can be viewed as the DFT with a few modifications. The relationship between the DFT and IDFT can be established as follows:

Start with DFT:

[ X[k] sum_{n0}^{N-1} x[n] expleft(-i frac{2pi}{N} knright) ]

Take the complex conjugate:

[ X[k]^* sum_{n0}^{N-1} x[n]^* expleft(i frac{2pi}{N} knright) ]

Switch the roles of ( x[n] ) and ( X[k] ):

The IFFT can be interpreted as swapping the roles of ( x[n] ) and ( X[k] ), which leads to:

[ x[n] frac{1}{N} sum_{k0}^{N-1} X[k] expleft(i frac{2pi}{N} knright) ]

4. Fast Fourier Transform (FFT)

The Fast Fourier Transform (FFT) is an efficient algorithm to compute the DFT. The IFFT can also be computed using the FFT algorithm with minor modifications. The key points are:

Input and Output: When using FFT to compute the IFFT the input is the frequency domain signal ( X[k] ) and the output will be the time domain signal ( x[n] ). Scaling Factor: The output of the FFT needs to be scaled by ( frac{1}{N} ) to account for the normalization in the IFFT.

Summary of IFFT Algorithm

To compute the IFFT, follow these steps:

Compute the FFT of the input sequence: ( X[k] ). Scale the result by ( frac{1}{N} ):

[ x[n] frac{1}{N} text{FFT}(X[k]) ]

Conclusion

The IFFT essentially reverses the process of the DFT while introducing a scaling factor. By utilizing the FFT algorithm, the computational efficiency of the IFFT can be significantly improved, allowing for fast calculations in various applications such as signal processing and data analysis.