Discrete Fourier Transform with different period

In summary, The problem at hand is how to efficiently evaluate sums of the form Y_k = \sum_{j=0}^{n-1} c_j e^{\frac{i j k \alpha}{n}}, where the value of \alpha may not be equal to 2\pi. This makes it difficult to use a standard DFT form and a regular FFT library. One solution is to use a windowing function to get rid of high frequency components and reduce blurring of the desired frequency components. There are various windowing functions available, such as the Hanning and Hamming window, and they are often built into signal processing programs like Matlab.
  • #1
vibe3
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Hi all, I have a seemingly simple problem which is I'd like to efficiently evaluate the following sums:

[tex]
Y_k = \sum_{j=0}^{n-1} c_j e^{\frac{i j k \alpha}{n}}
[/tex]

for [itex]k=0...n-1[/itex]. Now if [itex]\alpha = 2\pi[/itex], then this reduces to a standard DFT and I can use a standard FFT library to compute the sums. But if [itex]\alpha \ne 2\pi[/itex] then I don't see how I can put this into standard DFT form to use a regular FFT library on this.

I guess this problem amounts to computing a DFT with harmonics that do not necessarily have periods of [itex]2 \pi[/itex].

Any help is appreciated!
 
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  • #2
This situation happens all the time when you analyse measured data, because obviously you don't know exactly what frequencies are in the data till AFTER you have measured it and processsed it.

If you do a DFT on the raw data, the result will be hard to understand, because the discontinuity between the two ends of the sample will produce a lot of high frequency Fourier components that don't mean anything physically.

One standard technique is to multiply the data by a "windowing function" which is close to 0 at the ends of the range, and close to 1 in the middle. That gets rid of the meaningless high frequency Fourier components by making the ends of the sample "match up" (both become close to 0), but at the cost of blurring the frequency components you are interested in.

There are several different windowing functions that have been invented, with different tradeoffs between the amount of filtering and the amount of blurring, but if you are new to this I would recommend the Hanning window
http://en.wikipedia.org/wiki/Hann_function Note, there is also a Hamming window - be careful with the spelling!

This might be useful: http://www.tmworld.com/electronics-news/4383713/Windowing-Functions-Improve-FFT-Results-Part-I (and also part 2)

"Real world" signal processing programs like Matlab have these windowing options built in, so you don't have to work out the math yourself.
 

Related to Discrete Fourier Transform with different period

1. What is the Discrete Fourier Transform (DFT)?

The Discrete Fourier Transform is a mathematical tool used to convert a signal from its original domain (usually time or space) to a representation in the frequency domain. It is a discrete version of the Fourier Transform, which is a continuous mathematical operation used to decompose a function into its constituent frequencies.

2. How does the DFT handle signals with different periods?

The DFT can handle signals with different periods by using a technique known as zero-padding. This involves adding zeros to the end of the signal to make it a multiple of the original period. This allows the DFT to accurately represent the signal's frequency content and handle different periods.

3. What are the advantages of using the DFT for signals with different periods?

The DFT has several advantages for signals with different periods. Firstly, it allows for efficient computation and analysis of signals in the frequency domain. Additionally, it can handle non-periodic signals as well as signals with multiple periods simultaneously. This makes it a versatile tool for signal processing and analysis.

4. Are there any limitations to using the DFT for signals with different periods?

One limitation of using the DFT for signals with different periods is the potential for spectral leakage. This occurs when the signal's frequency components fall in between the discrete frequency bins of the DFT, resulting in a distorted representation in the frequency domain. This can be mitigated by using windowing techniques or increasing the DFT size.

5. How can the DFT be applied in real-world scenarios with signals of different periods?

The DFT has many real-world applications, such as in audio and image processing, telecommunications, and data compression. In these scenarios, the DFT can be used to analyze and manipulate signals with different periods, allowing for efficient and accurate processing of complex signals. It is also commonly used in scientific research for analyzing data from experiments and simulations.

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