A video takes you to understand the whole process of Fourier transform
Fourier series video analysis
Discrete Fourier Series ( DFS ) of Periodic Sequences
The periodic sequence does not satisfy the absolute summability condition, and its FT cannot be directly calculated by the definition formula
Discrete Fourier Series Transform Pair (DFS):
Four Fourier Transforms
Fourier Transform of Periodic Co(sine) Sequence
The FT of cos ω 0 n is the unit impulse function at ω = ± ω 0 of strength π and extended with a period of 2π
Some special sequences of DTFT
(1) DTFT with all 1s sequence
The FT of an all- ones sequence is a series of unit impulse functions centered at ω=0 and spaced at integer multiples of 2π, with an intensity of 2π and a period of 2π
The relationship between the Fourier transform of discrete signals in the time domain and the Fourier transform of analog signals
in conclusion:
1. The spectrum of the discrete signal in the time domain is also the periodic extension of the spectrum of the analog signal, and the period is
2. The FT of the analog signal can be calculated by calculating the FT of the corresponding time-domain discrete signal: First, according to the sampling theorem, the analog signal is sampled at a frequency more than twice the highest frequency of the analog signal to obtain the time-domain discrete signal, and then the computer is used. The time-domain discrete signal is subjected to FT to obtain its spectral function, and finally multiplied by the sampling interval Ts to obtain the FT of the analog signal. Note that the relationship on the frequency axis is ω = ΩT .
General Law of Fourier Transform
General rule: the discreteness of one domain will inevitably lead to the periodic extension of another domain
Definition of Discrete Fourier Transform
The relationship between DFT and Z transform, Fourier transform of sequence
The N -point DFT of the sequence x ( n ) is equivalent to taking N -point equally spaced sampling on the unit circle of the z -transform of x ( n ) , and the first sampling point should be taken at z = 1 .
X ( k ) is the Fourier transform of x ( n ) X ( e jω ) sampled at N points equally spaced on the interval [0 , 2π] .
Periodic Sequence and Periodic Extended Sequence
The second physical meaning of DFT (the relationship between DFT and DFS)
in conclusion:
The discrete Fourier transform X ( k ) of the finite-length sequence x(n) is exactly the principal value sequence of the discrete Fourier series coefficients of the periodic extension sequence x((n))N of x(n).
X ( k ) is essentially the spectral characteristics of the periodic extension sequence x((n))N of x(n), which is the second physical interpretation (physical meaning) of N-point DFT.
Linearity
Cyclic shift properties
Time Domain Cyclic Shift Theorem and Frequency Domain Cyclic Shift Theorem
Circular Convolution Theorem
The time-domain circular convolution theorem is the most important theorem in DFT and has strong practicality. Knowing the system input and the unit impulse response of the system, calculating the output of the system, and realizing the FIR filter with FFT , etc., are all based on this theorem.
Notice:
When the length L of the circular convolution interval is greater than or equal to the length of y(n) = h(n)*x(n), the circular convolution result is equal to the linear convolution.
Time Domain Circular Convolution Theorem
Frequency Domain Circular Convolution Theorem
