Rapid changes in signal correspond to high frequencies - proof

1. How does adding more higher frequency waves affect the resulting wave forms that you are trying to reproduce?
2. How do you shift a signal in the frequency domain?
3. Why do we convert signal from time domain to frequency domain?
4. What is frequency in Fourier transform?

How does adding more higher frequency waves affect the resulting wave forms that you are trying to reproduce?

As you add sine waves of increasingly higher frequency, the approximation gets better and better. The addition of higher frequencies better approximates the rapid changes, or details, in the original function. Because the waveform lacks symmetry, both even and odd harmonics are needed to approximate the function.

How do you shift a signal in the frequency domain?

Multiplying a function with a complex exponential with period t0=1/f0, corresponds to shifting the frequency domain by 1/t0=f0.

Why do we convert signal from time domain to frequency domain?

For mathematical systems governed by linear differential equations, a very important class of systems with many real-world applications, converting the description of the system from the time domain to a frequency domain converts the differential equations to algebraic equations, which are much easier to solve.

What is frequency in Fourier transform?

The term Fourier frequency most usually denotes the frequency of one of several components of a function which may or may not be periodic.