# How does ifft(fft(x)) form the correct bijection with domain?

• davidbenari
In summary, the conversation discusses the use of inverse fast-fourier transforms and fast-fourier transforms in approximating the continuous convolution of two signals. The speaker questions the reliability of this method and wonders how the resulting vector would correspond to the original domain. They also mention using MATLAB to compute the inverse transform of 1, which results in a dirac-delta, and express a need for further clarification and understanding of the theory behind this approach.
davidbenari
I think my question is more appropriate here than in the computation section. My question is:

(In the context of inverse fast-fourier transforms and fast-fourier transforms)

Knowing ifft(fft(x)=x might be trivial as it is almost a definition; associating it with a domain ##t## is perfectly fine since we were assuming ##x=x(t)## .

Now suppose I want to approximate the continuous convolution of two signals ##f,g##, both defined on ##t##. I could do it as CONV=ifft(fft(f).*fft(g)).

How can I be sure that the resulting vector will form the correct bijection with the same domain ##t##? In other words, why is it true that the first element of ##t## in fact does map to the first element of ##CONV## (what determines correctness here is what would happen if I had done things analytically)? I am currently thinking that by doing fast-fourier transforms you loose all information about domains, since computationally FFT's are defined to operate on vectors (which need no domain to begin with).

I think I'm right because if I use MATLAB to compute the inverse transform of 1 I get mapped to a dirac-delta but not to one that's centered at 0.

I was hoping someone might clarify this for me.

edit: I realize this might be a bad approximation to a continuous convolution. So if you have any better ideas please tell me. Discrete convolution doesn't seem to be what I'm looking for, according to what I've read. Specifically, I'm analyzing the response of an RC circuit with given input signals x(t)

Thanks.

Last edited:
This post could be deleted. I'm realizing I need to read a lot more about the theory and because of this my question really doesn't make sense.

## What is the purpose of using ifft(fft(x)) in a bijection with domain?

The purpose of using ifft(fft(x)) in a bijection with domain is to transform a signal from the time domain to the frequency domain, and then back to the time domain again. This process is commonly used in signal processing and allows for easier analysis and manipulation of the signal.

## How does ifft(fft(x)) ensure a bijection with the domain?

ifft(fft(x)) ensures a bijection with the domain by utilizing the inverse Fast Fourier Transform and the Fast Fourier Transform algorithms. These algorithms are designed to accurately map each point in the time domain to a corresponding point in the frequency domain and vice versa, ensuring a one-to-one mapping between the two domains.

## What is the mathematical explanation behind ifft(fft(x)) forming a bijection with the domain?

The mathematical explanation behind ifft(fft(x)) forming a bijection with the domain lies in the properties of the Fourier Transform. The Fourier Transform is a unitary transformation, meaning that it preserves the inner product between two signals. This property guarantees that the inverse transform, ifft, will also preserve the inner product, ensuring a bijection between the two domains.

## Are there any limitations to using ifft(fft(x)) for forming a bijection with the domain?

While ifft(fft(x)) is a commonly used method for forming a bijection with the domain, it does have some limitations. One limitation is that it assumes the signal is periodic, meaning that it repeats infinitely. This may not accurately reflect real-world signals, and alternative methods may need to be used in these cases.

## How does ifft(fft(x)) compare to other methods for forming a bijection with the domain?

ifft(fft(x)) is a popular method for forming a bijection with the domain due to its efficiency and accuracy. However, there are other methods, such as Wavelet Transforms and Z-Transforms, that can also be used for this purpose. The choice of method will depend on the specific characteristics and needs of the signal being analyzed.

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