Question about CSFT (2d FFT)

[TecHNooB]

CSFT = Continuous Surface Fourier Transform
2D FFT = Discrete Windowed Version

If the magnitude plot of a sine and cosine both occupy the 1st and 3rd quadrants, what do peaks in the 2nd and 4th quadrant represent? Usually I think of fourier transforms as a sum of sines and cosines, but clearly something is missing when I don't know what happens in the 2nd and 4th quadrants. Seems like all fourier signals can occupy the 1st and 3rd.

 

[schenley101]

fourier "series" are a sum of sine and cosines. the fourier transform is a the steady state response of the system

 

[toslat]

CSFT = Continous Space (not surface) Fourier Transform.

Continous Time Fourier Transfrom (CTFT) and Discrete Time Fourier Transform (DTFT) operate in two differnt domains. FFT is an algorithm for computing DTFT.

 

[CP5670]

I'm not sure what the question here is. Are you looking at the Fourier transform of a function of two variables?

Continous Time Fourier Transfrom (CTFT) and Discrete Time Fourier Transform (DTFT) operate in two differnt domains. FFT is an algorithm for computing DTFT.

FFT algorithms compute the DFT, which is finite. A DTFT is an inverse Fourier series. All of these are just different versions of the same concept though.

 

[TecHNooB]

My question is what the peaks in the 2nd and 4th quadrant of a CSFT represent. I am looking for insight on the FT of a 2-D function (specifically the meaning of a peak in the 2nd or 4th quadrant).

Example: f(x,y) = cos(2*pi*(x*u0 + y*v0))

If I plot the magnitude of F(u,v) (CSFT of f(x,y)), I get an impulse in the first and third quadrant corresponding to the frequencies u0 and v0. I want to know how I obtain peaks in the 2nd and 4th and what they mean.

 

[CP5670]

Just set the u0 and v0 to have different signs and you'll get peaks in the other quadrants instead.

 

[toslat]

FFT algorithms compute the DFT, which is finite. A DTFT is an inverse Fourier series. All of these are just different versions of the same concept though.

While the expressions are similar, a DTFT is not an inverse Fourier series, as a Fourier series is based on continuous time while a DTFT is a forward transform on discrete time. In addition, for DTFT, the frequency range is bounded by f= [-1/2, 1/2] or a similar range.

A DFT is a DTFT with finite length N and discrete frequency f (which are implicitly required for any computerized implementations).

In truth, you cannot be bounded in both frequency and time.

 

[toslat]

@OP
exp(2*pi*x*u0)*exp(2*pi*x*v0) will give you an impulse at (u0, v0) or at its image, depending on how you define your foward and inverse transforms. Changing u0 and/or v0 will move the impulse accordingly in the (u,v) plane.

Interpretation of the location of an impulse will depend on the area of application.

 

[CP5670]

While the expressions are similar, a DTFT is not an inverse Fourier series, as a Fourier series is based on continuous time while a DTFT is a forward transform on discrete time. In addition, for DTFT, the frequency range is bounded by f= [-1/2, 1/2] or a similar range.

That's just a matter of what you interpret to be time or frequency. Mathematically, they are simply the inverse operators of each other.

 

[toslat]

That's just a matter of what you interpret to be time or frequency. Mathematically, they are simply the inverse operators of each other.

its not a matter of interpretation or choice of domain. For example, an essential part of defining the fourier series equivalent expression is the associated fundamental frequency, which is not relevant in the case of the DTFT as the range is normalized.

If one ignores such differences, one can generalize the whole of Fourier analysis into the use of exponential basis functions.

 

[CP5670]

its not a matter of interpretation or choice of domain. For example, an essential part of defining the fourier series equivalent expression is the associated fundamental frequency, which is not relevant in the case of the DTFT as the range is normalized.

The fundamental frequency is related to the size of the interval the function is defined on, which is a matter of normalizing things and has no significance beyond that. If you're dealing with two finite intervals of different sizes, you can easily map functions from one to the other.

Look at it like this. The Fourier series operation is a bijective map from L2(T) to l2 (it takes functions on a circle or interval and produces sequences). The DTFT is the inverse of that operator (it takes sequences and produces functions).

If one ignores such differences, one can generalize the whole of Fourier analysis into the use of exponential basis functions.

Not sure what you're trying to say here, but you can indeed extend most of the basic theory to a lot of other situations.

 

[toslat]

The fundamental frequency is related to the size of the interval the function is defined on, which is a matter of normalizing things and has no significance beyond that. If you're dealing with two finite intervals of different sizes, you can easily map functions from one to the other.

Look at it like this. The Fourier series operation is a bijective map from L2(T) to l2 (it takes functions on a circle or interval and produces sequences). The DTFT is the inverse of that operator (it takes sequences and produces functions).

There is a difference between a sequence of numbers, and samples of a function, cos while both can be represented by a set of numbers, the sampling interval is relevant in one case and not the other.

If I give you a finite set of of numbers, you can compute the DTFT with it, but you cannot use it alone as fourier series coefficients, as the 'normalization' you are trivializing is needed. sin(2x) is not sin(x); similarity does not mean equality.

Not sure what you're trying to say here, but you can indeed extend most of the basic theory to a lot of other situations.

Fourier analysis in the general sense is an expansion on exponential basis functions. The mode of summation, basis of support, domain etc are what makes the subtle differences.

 

[CP5670]

There is a difference between a sequence of numbers, and samples of a function, cos while both can be represented by a set of numbers, the sampling interval is relevant in one case and not the other.

If I give you a finite set of of numbers, you can compute the DTFT with it, but you cannot use it alone as fourier series coefficients, as the 'normalization' you are trivializing is needed. sin(2x) is not sin(x); similarity does not mean equality.

The point is that the concept is the same. The set of all L2 functions that correspond to that sequence are all scaled or shifted versions of each other.

What you're claiming is that rescaling leads to a different transform, which is like saying that the various normalizations of the (continuous) Fourier transform in use out there are different transforms. Sure, they are technically different, but conceptually identical.

Fourier analysis in the general sense is an expansion on exponential basis functions. The mode of summation, basis of support, domain etc are what makes the subtle differences.

If you mean Fourier analysis as a subject, it covers a lot more than just that.

The various kinds of Fourier transforms/expansions aren't really different in a sense, and can be put into a single, general framework. This is why most results in the discrete-time and continuous-time cases are very similar.

 

[TecHNooB]

Btw, the answer I was looking for was rotation, which is definitely something new when we move to the 2 dimensions.

https://engineering.purdue.edu/~bouman/ece637/notes/pdf/CSFT.pdf

^ Incredibly useful