# Fourier transform vs Inner product

• Bipolarity
In summary, the complex exponential Fourier series form an orthonormal basis for the space of functions, with a periodic function requiring countably many elements and an aperiodic function requiring uncountably many elements. The coefficients of the exponentials can be found using either the Fourier transform or the inner product with a complex exponential, although there may be some differences between the two formulas. Additionally, it is important to clarify whether discussing Fourier series or Fourier transforms, as there are different conventions in use.
Bipolarity
So the complex exponential Fourier series form an orthonormal basis for the space of functions. A periodic function can be represented with countably many elements from the basis, and an aperiodic function requires uncountably many elements.

Given a signal, we can find the coefficients of the exponentials in two ways:
1) Fourier transform
2) Inner product with that complex exponential

Though these two formulas are similar, they are not identical. So how could they both possibly give us the coefficient of a complex exponential?

Thanks!

BiP

Can you please show the formulas you are comparing? There are several different conventions in use. Also, please clarify whether you are talking about Fourier series or Fourier transforms. You mentioned both.

[quotr]A periodic function can be represented with countably many elements from the basis, and an aperiodic function requires uncountably many elements.
[/quote]
What do have in mind? The basis has only a countable number of elements. Are you mixing Fourier series and Fourier transdforms?

an aperiodic function requires uncountably many elements.
But not necessarily uncountably many non-zero elements. For example ##\cos t + \cos \pi t##.

But I agree with the other posters, it's hard to figure out exactly what your OP is asking.

olarBear, thank you for your question. The Fourier transform and inner product are two different mathematical techniques used to analyze signals, specifically in the context of the Fourier series. While they may seem similar, they are fundamentally different and have their own unique applications.

The Fourier transform is a mathematical operation that decomposes a signal into its constituent frequencies. It takes a signal in the time domain and converts it into the frequency domain, where the amplitude and phase of each frequency component can be analyzed. This is useful for analyzing periodic signals, as it allows us to see the individual frequencies that make up the signal.

On the other hand, the inner product is a mathematical concept that measures the similarity between two vectors. In the context of the Fourier series, the inner product is used to find the coefficients of the complex exponential basis functions that make up a signal. It involves taking the integral of the product of the signal and the complex exponential over a certain interval. This is useful for analyzing both periodic and aperiodic signals, as it allows us to find the coefficients for any type of signal.

So, how can both the Fourier transform and inner product give us the coefficients of a complex exponential? The answer lies in the fact that the complex exponential functions form an orthonormal basis for the space of functions. This means that they are orthogonal (perpendicular) to each other and have a unit norm. This property allows us to use both the Fourier transform and inner product to find the coefficients of the complex exponentials in a signal.

In summary, while both the Fourier transform and inner product can be used to find the coefficients of complex exponential functions, they are different techniques with their own unique applications. Understanding the differences between them is important for accurately analyzing and interpreting signals in the Fourier series.

## 1. What is the difference between Fourier transform and inner product?

The Fourier transform is a mathematical operation that decomposes a signal into its frequency components, while the inner product is a mathematical operation that measures the similarity between two signals.

## 2. How are Fourier transform and inner product related?

The Fourier transform can be expressed as an inner product between the signal and a set of basis functions, known as the Fourier basis. In this sense, the Fourier transform is a special case of the inner product.

## 3. What are the applications of Fourier transform and inner product?

The Fourier transform is commonly used in signal processing, image processing, and data compression. The inner product has applications in fields such as linear algebra, quantum mechanics, and optimization.

## 4. Can Fourier transform and inner product be used interchangeably?

No, they serve different purposes and cannot be used interchangeably. The Fourier transform is used for analyzing signals in the frequency domain, while the inner product is used for measuring similarity and performing calculations in vector spaces.

## 5. Are there any limitations to using Fourier transform and inner product?

Both Fourier transform and inner product have limitations depending on the specific application. For example, the Fourier transform may not be applicable for signals with discontinuities or infinite energy. The inner product may not be suitable for non-linear problems or when the signals are not orthogonal.

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