Fourier transform vs Inner product

[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

 

[jbunniii]

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.

 

[mathman]

[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?

 

[AlephZero]

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.

 

[blue_raver22]

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.