ABSTRACT
The essential usefulness of the DFT is in its ability to approximate the continuous Fourier transform (CFT).
Discretization Intervals
T= 2 π ω s https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780429332951/a5e9ddd2-7a84-42d5-9c66-2199fdb5b813/content/eq477.tif"/>
Time domain:
Ω 0 = 2 π T 0 https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780429332951/a5e9ddd2-7a84-42d5-9c66-2199fdb5b813/content/eq478.tif"/>
Frequency domain:
W N = e j 2 π N https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780429332951/a5e9ddd2-7a84-42d5-9c66-2199fdb5b813/content/eq479.tif"/>
N roots of unity:
T 0 = NT https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780429332951/a5e9ddd2-7a84-42d5-9c66-2199fdb5b813/content/eq480.tif"/>
Truncation Intervals
Ω 0 = N Ω 0 https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780429332951/a5e9ddd2-7a84-42d5-9c66-2199fdb5b813/content/eq481.tif"/>
Time domain:
T0 = NT
Frequency domain:
Ω0 = NΩ0
N roots of unity: Synthesis equation:
W N − 1 = e − j 2 π N https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780429332951/a5e9ddd2-7a84-42d5-9c66-2199fdb5b813/content/eq482.tif"/>
() x ( nT ) = 1 NT ∑ k=0 N-1 X ( k Ω 0 ) W N kn = Ω 0 2 π ∑ k=0 N-1 X ( k Ω 0 ) W N kn , 0 ≤ n< N 2 https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780429332951/a5e9ddd2-7a84-42d5-9c66-2199fdb5b813/content/eq483.tif"/> () X ( Ω 0 ) = T ∑ n=0 N-1 x ( nT ) W − kn = T 0 N ∑ n=0 N-1 x ( nT ) W − kn , 0 ≤ n< N 2 https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780429332951/a5e9ddd2-7a84-42d5-9c66-2199fdb5b813/content/eq484.tif"/>Time Functions
122A semi-infinite time function x(t) is given. It is necessary to discretize this function so that the discrete Fourier transform can be determined.
The sampling function y1(t) is a comb function with impulses of strength T and sampling interval T. T is chosen for acceptable aliasing error. () y 1 ( t ) = T ∑ n = − ∞ ∞ δ ( t − nT ) https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780429332951/a5e9ddd2-7a84-42d5-9c66-2199fdb5b813/content/eq485.tif"/>
xs(t) = x(t)y1(t) has to be truncated in the time domain at a suitably determined time interval T0. The sampled function xs(t) can be written as: () x s ( t ) = x ( t ) y 1 ( t ) = T ∑ n = − ∞ ∞ x ( nT ) δ ( t − nT ) https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780429332951/a5e9ddd2-7a84-42d5-9c66-2199fdb5b813/content/eq486.tif"/>
The truncation filter w(t) is an ideal lp filter, pT 0 2 ( t ) https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780429332951/a5e9ddd2-7a84-42d5-9c66-2199fdb5b813/content/eq487.tif"/> , offset slightly by −ε so that only (N − 1) samples are included. The truncation time T0 is chosen to satisfy the frequency resolution Ω 0 = 2 π T 0 https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780429332951/a5e9ddd2-7a84-42d5-9c66-2199fdb5b813/content/eq488.tif"/> T0 = NT. () w ( t ) = pT 0 / 2 ( t − T 0 2 + ε ) https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780429332951/a5e9ddd2-7a84-42d5-9c66-2199fdb5b813/content/eq489.tif"/>
The truncated sampled function X̃s(t) is given by: () x ˜ s ( t ) = x s ( t ) w ( t ) = T ∑ n = − ∞ N − 1 x ( nT ) δ ( t − nT ) https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780429332951/a5e9ddd2-7a84-42d5-9c66-2199fdb5b813/content/eq490.tif"/>
https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780429332951/a5e9ddd2-7a84-42d5-9c66-2199fdb5b813/content/ufig22_1.tif"/>123 124The corresponding time function y2(t) is again another comb function with spacing T0. () y 2 ( t ) = ∑ n= − ∞ ∞ δ ( t − kT 0 ) : T 0 = 2 π Ω 0 https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780429332951/a5e9ddd2-7a84-42d5-9c66-2199fdb5b813/content/eq491.tif"/>
The discrete time periodic sequence x̃sp(t) is: () x ˜ sp ( t ) = x ˜ ( t ) y 2 ( t ) = T ∑ k = − ∞ ∞ [ ∑ n = 0 N − 1 x ( nT ) δ ( t − nT-kT 0 ) ] https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780429332951/a5e9ddd2-7a84-42d5-9c66-2199fdb5b813/content/eq492.tif"/>
Note that Ω 0 T= 2 π N https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780429332951/a5e9ddd2-7a84-42d5-9c66-2199fdb5b813/content/eq493.tif"/> so that T= 2 π N Ω 0 https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780429332951/a5e9ddd2-7a84-42d5-9c66-2199fdb5b813/content/eq494.tif"/>
https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780429332951/a5e9ddd2-7a84-42d5-9c66-2199fdb5b813/content/ufig22_2.tif"/>Frequency Functions
125 126The corresponding continuous time Fourier transform, X(ω) is also known.
The Fourier transform Y1 ω) is also a comb function given by: () Y 1 ( ω ) = 2 π ∑ n = − ∞ δ ( ω − nω s ) : ω s = 2 π T https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780429332951/a5e9ddd2-7a84-42d5-9c66-2199fdb5b813/content/eq495.tif"/>
The Fourier transform Xs(ω) is a continuous periodic function with period ω s = 2 π T https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780429332951/a5e9ddd2-7a84-42d5-9c66-2199fdb5b813/content/eq496.tif"/> . T is so chosen that the aliasing error is within acceptable limits. () X s ( ω ) = 1 2 π X ( ω ) * Y 1 ( ω ) = ∑ n = − ∞ X ( ω − nω s ) https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780429332951/a5e9ddd2-7a84-42d5-9c66-2199fdb5b813/content/eq497.tif"/>
The Fourier transform W(ω) is a Sa(ω) function with zero crossings at the desired frequency resolution Ω 0 = 2 π T 0 https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780429332951/a5e9ddd2-7a84-42d5-9c66-2199fdb5b813/content/eq498.tif"/> . () W ( ω ) = T 0 e − jω ( T 0 / 2 − ε ) Sa ( ωT 0 2 ) https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780429332951/a5e9ddd2-7a84-42d5-9c66-2199fdb5b813/content/eq499.tif"/>
The corresponding FT, X̃s(ω) is obtained by directly taking the FT of x̃s(t). X̃s(ω) is still a continuous periodic function that has to be discretized by multiplying with a comb function Y2(ω). () X ˜ s ( ω ) = X s ( ω ) * W ( ω ) = T ∑ n = 0 N − 1 x ( nT ) e − jωnT https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780429332951/a5e9ddd2-7a84-42d5-9c66-2199fdb5b813/content/eq500.tif"/>
https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780429332951/a5e9ddd2-7a84-42d5-9c66-2199fdb5b813/content/ufig22_3.tif"/>127 128The sampling function Y2(ω) is a comb function with sampling interval Ω0 set at the desired frequency resolution and impulse strength Ω0. () Y 2 ( ω ) = Ω 0 ∑ n = − ∞ ∞ δ ( ω − k Ω 0 ) https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780429332951/a5e9ddd2-7a84-42d5-9c66-2199fdb5b813/content/eq501.tif"/>
The discrete frequency periodic sequence X̃sp(ω) is () X ˜ sp ( ω ) = X ˜ s ( ω ) * Y 2 ( ω ) = Ω 0 ∑ n = 0 N − 1 X s ( k Ω 0 ) δ ( ω-k Ω 0 ) https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780429332951/a5e9ddd2-7a84-42d5-9c66-2199fdb5b813/content/eq502.tif"/> Ω 0 = 2 π NT https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780429332951/a5e9ddd2-7a84-42d5-9c66-2199fdb5b813/content/eq503.tif"/>
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