DFT
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Draft notes for Section 27 of a spectral theory book, marked as on probation. It compares the Fourier integral transform with a digital (sampled-time) transform and questions whether the latter implies a pulse train. It then proposes a discrete transform for a finite pulse fitting in (0,T1) with N samples and verifies the inversion using the sum of exponentials giving a Kronecker delta. The text is cut off mid-section and some equations are garbled.
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27. The Discrete Fourier Transform
This entire section is on probation for a while.
First, we consider the Discrete Fourier Transform
Fourier Integral Transform (both sides are integrals)
X(ω) = !Syntax Error, Idt x(t) e-iωt projection = transform (1.1)
x(t) = (1/2π) !Syntax Error, Idω X(ω) e+iωt expansion = inverse transform (1.2)
Digital Fourier Transform ( spectrum is a sum, expansion is an integral)
X'(ω) ≡ !Syntax Error, I∆t x(tn) e-iωt projection = transform (22.2)
x(tn) = !Syntax Error, I dω X'(ω) e+iωt expansion = inversion (22.4)
Is this a pulse or a pulse train? Question: Suppose we apply 22.2 to signal which happens to be finite. Look at the derivation of the above and what then happens? Something is fishy!
Here is a web site that is not to clean , but here is one conclusion
http://fourier.eng.hmc.edu/e101/lectures/handout4/node3.html
Here you see discrete time x[m] given as a one-period integral, as in mine, and you see the spectrum as a sum just as in mine. So this is my Digital Fourier Transform. The time domain is discrete steps, but the ω domain is a sum of continuous functions, so continuous. OK.
When I do my Digital FT Section 22, I never mention a pulse train and I never mention Xpulse.
(a) Discrete Fourier Transform of a Finite Pulse x(t)
I conjecture that this is the correct transform
x(tn) = !Syntax Error, Ic'm e+imn(2π/N) n = 0,1....N-1
c'm ≡ (1/T1) !Syntax Error, I ∆t x(tn) e-imωt m = 0,1,....N-1
∆t = T1/N T1 = N ∆t ω1 = (2π/T1) = 2π/(N∆t) tn = n∆t (27.3)
The entire pulse is assumed to fit inside the interval (0,T1). In this case,
tn lies in (0,nmaxΔt) = (0,NT1/N) = (0,T1 ) check
So index n runs from n = 0 to n = N-1 I suspect.
Now let's verify this transform as in spectral doc:
x(tn) = !Syntax Error, Ic'm e+imn(2π/N) = !Syntax Error, I[(1/N) !Syntax Error, Ix(tk) e-imk(2π/N)] e+imn(2π/N)
= (1/N) !Syntax Error, Ix(tk) !Syntax Error, I e+im(2π/N)(n-k) (27.12)
= (1/N) !Syntax Error, Ix(tk) N !Syntax Error, Iδk,n-mN = !Syntax Error, I !Syntax Error, Ix(tk) δk,n-mN
= x(tn-mN) = x(tn-mT1) . (27.14)
= 1/N) !Syntax Error, Ix(tk) N !Syntax Error, Iδk,n-mN
= (1/N) !Syntax Error, Ix(tk) N δk,n = x(tn)
(a) Discrete Fourier Transform of a Finite Pulse x(t) of width T1
Recall the