Transform_1

 Before looking at the Quantum Fourier Transform, decided to get clear about the Fourier 

Transform within mathematics. It's a complicated matter. Did pick up a few interesting things 

on Day 1 of this problem.

Microsoft Word - Fourier Analysis Notes.doc (stanford.edu)                          

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COPILOT

Fourier analysis; how can a non periodic function exhibit frequency?

Great question! Fourier analysis can indeed be applied to non-periodic functions to analyze their frequency components. This is done using the Fourier transform, which is a more general tool compared to the Fourier series.

Here’s a simplified explanation:

Fourier Series vs. Fourier Transform:

Fourier Series is used for periodic functions and represents them as a sum of sine and cosine functions with discrete frequencies.

Fourier Transform extends this concept to non-periodic functions, transforming them into a continuous spectrum of frequencies.

Frequency Representation:

Even if a function is not periodic, it can still be decomposed into a sum of sine and cosine functions with different frequencies. The Fourier transform provides a way to represent any function in terms of its frequency components.

Mathematical Insight:

For a non-periodic function ( f(t) ), the Fourier transform ( F(\omega) ) is given by:F(ω)=∫−∞∞​f(t)e−iωtdt

This integral transforms the time-domain function ( f(t) ) into a frequency-domain function ( F(\omega) ), where ( \omega ) represents the angular frequency.

Practical Applications:

This approach is widely used in signal processing, physics, and engineering to analyze the frequency content of signals that do not repeat over time1

By using the Fourier transform, we can understand and analyze the frequency characteristics of non-periodic functions, which is crucial for various scientific and engineering applications.

Does this help clarify things? Feel free to ask if you have more questions!

Fourier analysis; what is the difference between frequency and period?

In the context of Fourier analysis and waveforms, frequency and period are two fundamental concepts that are closely related but distinct:

• Frequency refers to the number of complete cycles of a waveform that occur in one second. It is measured in Hertz (Hz). For example, if a wave completes 5 cycles in one second, its frequency is 5 Hz.

• Period is the duration of time it takes for one complete cycle of the waveform to occur. It is measured in seconds (s). For instance, if one cycle of a wave takes 0.2 seconds, the period is 0.2 s.

The relationship between frequency ((f)) and period ((T)) is given by the formula:

f=T1​

This means that frequency is the reciprocal of the period and vice versa¹². 

In simpler terms, if the frequency of a wave increases, the period decreases, and if the frequency decreases, the period increases. This inverse relationship is crucial in understanding wave behavior in Fourier analysis.

Does this help clarify the difference for you? If you have more questions about Fourier analysis or anything else, feel free to ask!

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Also discovered how-to create an orthonormal basis between any two vectors. Neat!!

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