文档介绍：Chapter 5 Finite-Length discrete Transforms
Definition of DFT
The relationship between DFT and DTFT
DFT Properties
putation
Discrete Fourier Transform (DFT)
Time domain Frequency domain
Continuous Aperiodical FT Continue Aperiodical
Continuous Periodical FS Discrete Aperiodical
Discrete Aperiodical DTFT Periodical Continuous
Discrete Periodical DFS Periodical Discrete
Typical DFT Pair
δT(t) ω0δω0(ω)
T
2T
-T
-2T
δT(t)
t
0
ω0
2ω0
- ω0
-2ω0
ω0δω0(ω)
0
ω
ω0 = 2π/T
Make a signal discrete and periodical
The engineering signals are often continuous and aperiodical. If we want to process the signals with DFT, we have to make the signals discrete and periodical.
Sampling to make the signal be discrete.
Make the signal periodical by periodical expanding.
Make a signal discrete and periodical
From DTFT to get DFT
DTFT: discrete in time domain, continuous in frequency domain.
Sampling the DTFT of sequences to get N frequency points to research, that is DFT.
1
2
3
4
5
6
7(k=N-1)
k=0
Re[z]
The Discrete Fourier Transform(DFT)
A kind of useful tool to process finite sequences.
Based on DFT ,we can puter to perform signal processing.
FFT algorithm is a core in DSP.
DFT is the same as the DFS.
The Definition of DFT
Where:
Which can easily be deduced from:
The Definition of DFT
Note: X[k] is also a length-N sequence in the frequency domain.
The sequence X[k] is called the Discrete Fourier Transform (DFT) of the sequence x[n].
To verify the above expression we multiply both sides of the above equation by WNln and sum the result from n = 0 to n=N-1.
The Definition of DFT
resulting in: