# Fourier Transform: best window to represent function

• Engineering
• Master1022
In summary: For example, the ramp function is the result of convolving the top-hat function with itself. This is important because it reduces the amount of peripheral pulses (due to the sinc function being squared). Additionally, the ramp function is the result of convolving the top-hat function with itself, which means that it has a better noise-to-signal ratio (due to the reduced number of peripheral pulses).
Master1022
Homework Statement
We have a time-continuous signal $f(t)$. A new signal $g(t)$ is created by either by multiplying $f(t)$ with a top-hat function (half-width $\frac{T}{2}$) or a ramp function (half-width $T$), both with amplitude 1. Which window should we, using qualitative judgement, choose to have a better representation of $F(\omega )$
Relevant Equations
Fourier transform
Hi,

I was hoping to gain more insight into these window questions when looking at frequency spectra questions. I don't really know what makes windows better than one another.

My attempt:
In the question, we have $f(t) = cos(\omega_0 t)$ and therefore its F.T is $F(\omega ) = \pi \left( \delta(\omega - \omega_0 ) + \delta(\omega + \omega_0) \right)$. For the window functions, we have a top-hat function with a transform of:
$$\frac{ T sin(\omega T / 2)}{\omega T / 2}$$ and a ramp function with transform:
$$\frac{ 4 sin^2 (\omega T / 2)}{\omega^2 T}$$

To find the effect of multiplying the time signals, we can carry out convolution in the time domain and utilize the sifting property of the delta function.

I can see that we basically have the choice of $sinc$ or $sinc^2$. Perhaps the ramp function will be better as it has smaller peripheral pulses (due to the sinc function being squared). Also, I notice that the ramp function is the result of convolving the top-hat function with itself.

I am not sure what other aspects I should be looking out for.

Any help is greatly appreciated.

Master1022 said:
I was hoping to gain more insight into these window questions when looking at frequency spectra questions. I don't really know what makes windows better than one another.
Here is some information on various Windowing Functions from the PicoScope USB Oscilloscope Manual (I'm using one right now in one of my test setups at work to do FFTs and frequency domain analysis of powerline communication network waveforms). It's a good brief summary of Windowing functions, and should give you some good search terms for further searching/reading:

Master1022
Data analysis types each have their favorites, which means that none of them are the best. They each have their own pros and cons. Ideally, if you care, you will need to analyze the effects of each and pick the best for your particular analysis. The answer lies in being clear about what features of your data set you really care about most.

Master1022 and berkeman
@berkeman and @DaveE - thank you for your replies! There was more nuance to the choice than I previously thought.

DaveE and berkeman

## 1. What is a Fourier Transform?

A Fourier Transform is a mathematical tool used to analyze the frequency components of a given signal or function. It converts a function from the time or spatial domain to the frequency domain, allowing for a better understanding of the underlying patterns and behaviors of the function.

## 2. How does the Fourier Transform work?

The Fourier Transform decomposes a function into its constituent frequencies by representing it as a combination of sinusoidal functions. This is achieved through the use of complex numbers and integration over an infinite interval.

## 3. What is the best window to use for representing a function with the Fourier Transform?

The best window to use for representing a function with the Fourier Transform depends on the specific characteristics of the function being analyzed. Some commonly used windows include the rectangular, Hamming, and Blackman windows, each with its own advantages and limitations.

## 4. How do I choose the best window for my function?

The best window for a specific function can be chosen by considering factors such as the frequency resolution and spectral leakage of the window. It is also helpful to experiment with different windows and compare the resulting Fourier Transforms to determine which one best represents the function.

## 5. Are there any alternatives to using a window with the Fourier Transform?

Yes, there are alternative methods for representing a function with the Fourier Transform, such as using a Gaussian window or applying a zero-padding technique. These methods may be more suitable for certain types of functions or for specific analysis purposes.

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