Understanding the Inverse Fast Fourier Transform (IFFT) Function in R

In this article, we’ll delve into the world of Fast Fourier Transforms (FFTs), specifically focusing on the IFFT function and its implementation in R. We’ll explore how to replicate the behavior of Matlab’s ifft function using R’s built-in mvfft function with some clever data manipulation.

Introduction to FFTs and IFFTs

Fast Fourier Transforms are a class of algorithms that efficiently compute the discrete Fourier transform (DFT) of a sequence. The DFT is a mathematical operation that decomposes a function or a sequence of values into its constituent frequencies. Inverse Fast Fourier Transforms, on the other hand, perform the inverse operation: they take the frequency domain representation and convert it back to the time domain.

The FFT is based on the Cooley-Tukey algorithm, which divides the DFT into smaller sub-problems that can be solved recursively. This leads to an efficient computational cost of O(n log n), making FFTs a crucial tool in many fields, including signal processing, image analysis, and numerical analysis.

Matlab’s ifft Function

In Matlab, the ifft function is used to compute the inverse DFT of a sequence. The syntax for ifft is:

ifft(X, n, dim)

Where:

• X is the input sequence.
• n is the length of the sequence (not required if omitted).
• dim specifies which dimension to perform the inverse DFT across.

R’s mvfft Function

In R, the mvfft function from the signal package performs a multi-dimensional FFT. The syntax for mvfft is:

mvfft(x, inverse = FALSE)

Where:

• x is the input sequence.
• inverse is a logical vector indicating which dimensions to compute the inverse DFT across.

Replicating Matlab’s ifft Function in R

To replicate Matlab’s ifft function using R, we can use the mvfft function with some clever data manipulation. Let’s break this down step by step:

Reshaping Data for IFFT

R’s mvfft function performs the FFT along columns (by default). However, Matlab’s ifft function computes the inverse DFT across a specified dimension. To replicate this behavior in R, we need to reshape our data.

There are two main functions that can be used for reshaping data:

• t(): Transposes a matrix.
• aperm(): Permits an array by swapping dimensions.

We’ll use both of these functions to manipulate our input sequence.

Applying mvfft with Inverse = TRUE

To perform the inverse DFT, we need to set the inverse argument in mvfft to TRUE.

ifft_data <- mvfft(x, inverse = TRUE)

However, this will perform the FFT along columns (by default). To replicate Matlab’s behavior, we need to reshape our data.

Reshaping and Applying IFFT

To apply IFFT using mvfft, we need to transpose our input sequence (x) before applying mvfft. This will ensure that R performs the inverse DFT across all dimensions:

# Transpose x
ifft_data <- mvfft(t(x), inverse = TRUE)

Alternatively, if we want to preserve the original order of the data but still apply IFFT along the desired dimension, we can use aperm to swap the dimension and then transpose it before applying mvfft:

# Swap dimensions using aperm
ifft_data <- mvfft(aperm(x, 2), inverse = TRUE)

Note that this approach assumes that you know the exact position of the dimension you want to apply IFFT across. In Matlab’s ifft function, this is specified by the third argument (dim).

However, since R does not support specifying an arbitrary dimension for the inverse FFT directly like Matlab, we can use arrayhelpers’ convenience functions to convert between arrays and matrices.

For example, let’s assume that our input sequence x has a specific shape (e.g., 100 x 1). To reshape it into a matrix (i.e., dim <- -) where we want to apply IFFT across the first dimension:

# Convert array to matrix using dim<- 
ifft_data <- mvfft(as.matrix(x), inverse = TRUE)

Alternatively, if you need to convert from an array to matrix with specific dimensions after already performing the FFT using mvfft:

# Use convenience functions for reshaping and converting between arrays and matrices
library(arrayhelpers)

# Convert the result of mvfft back into a numeric vector 
ifft_data <- as.numeric(as.matrix(mvfft(x, inverse = TRUE)))

Note that this approach assumes that you know the shape of your input sequence.

Conclusion

Replicating Matlab’s ifft function in R requires some clever data manipulation using mvfft, t(), and aperm. By reshaping our data correctly and applying mvfft with the inverse argument set to TRUE, we can obtain the desired result.

Last modified on 2023-11-28