Peak of Analytical Fourier Transform

In summary, differentiating and setting to 0 is a general method for finding the peak frequency in a Fourier transform, but it cannot be applied in cases where the Fourier transform involves a Dirac delta function. In these special cases, other methods such as using computer software may be necessary to find the peak frequency.
  • #1
Luke Tan
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TL;DR Summary
Finding the peak frequency in an analytical fourier transform
In a numerical Fourier transform, we find the frequency that maximizes the value of the Fourier transform.

However, let us consider an analytical Fourier transform, of ##\sin\Omega t##. It's Fourier transform is given by
$$-i\pi\delta(\Omega-\omega)+i\pi\delta(\omega+\Omega)$$
Normally, to find the value of ##\omega## that maximizes this function, we would differentiate with respect to ##\omega## and set to 0. However, in this case, the derivative of a dirac delta function cannot be evaluated. Hence, we are unable to find the peak frequency to be at ##\Omega## unlike what we would in a numerical, discrete-time Fourier transform.

Is there any way around this, to find the peak frequency of the Fourier transform of a function?
 
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  • #2
Te Dirac delta is 0 everywhere except for ##\delta(0) \neq 0##. It shouldn't be too har to find where the FT has a peak :smile:
 
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  • #3
DrClaude said:
Te Dirac delta is 0 everywhere except for ##\delta(0) \neq 0##. It shouldn't be too har to find where the FT has a peak :smile:
oh so this is a special case, and does not disqualify differentiating and setting to 0 as a general method of finding the peak frequency?
 
  • #4
Luke Tan said:
oh so this is a special case, and does not disqualify differentiating and setting to 0 as a general method of finding the peak frequency?
Yes, it's a special case because the Dirac delta is not a real function, but a distribution, so you can't apply the same methods as you would normally use.
 
  • #5
What frequency is the sine wave oscillating at? That is where your delta function will be and at ##2 \pi - \Omega##

You could always use Matlab or etc. to find maxes and mins of Fourier's.
 

Related to Peak of Analytical Fourier Transform

1. What is the "Peak of Analytical Fourier Transform"?

The "Peak of Analytical Fourier Transform" is a mathematical concept used in signal processing and analysis. It refers to the highest point or value in the Fourier transform of a signal, which represents the dominant frequency component in the signal.

2. How is the "Peak of Analytical Fourier Transform" calculated?

The "Peak of Analytical Fourier Transform" is calculated by taking the Fourier transform of a signal and finding the frequency component with the highest amplitude or magnitude. This can be done using mathematical formulas or through digital signal processing algorithms.

3. What is the significance of the "Peak of Analytical Fourier Transform" in signal processing?

The "Peak of Analytical Fourier Transform" is significant because it allows us to identify the dominant frequency component in a signal, which can help in understanding the underlying patterns and characteristics of the signal. It is also used in filtering and noise reduction techniques.

4. Can the "Peak of Analytical Fourier Transform" be used for any type of signal?

Yes, the "Peak of Analytical Fourier Transform" can be used for any type of signal, including continuous and discrete signals. It is a fundamental concept in signal processing and is applicable to a wide range of signals in various fields such as engineering, physics, and biology.

5. How does the "Peak of Analytical Fourier Transform" differ from the "Peak of Discrete Fourier Transform"?

The "Peak of Analytical Fourier Transform" refers to the highest point in the Fourier transform of a continuous signal, while the "Peak of Discrete Fourier Transform" refers to the highest point in the Fourier transform of a discrete signal. The former is calculated using mathematical integration, while the latter is calculated using mathematical summation. However, both peaks represent the dominant frequency component in a signal.

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