EECS 451 DISCRETE-TIME FOURIER TRANSFORM
DTFT: X(ejω ) = P∞
n=−∞ x(n)e−jωn =X(z)|z=ej ω (z-xform on unit circle).
Inverse: x(n) = 1
2πRπ
−πX(ejω )ejωn dω. (−π, π)→(p−π, p +π) for any p.
Period: X(ejω ) is periodic with period 2π. Highest frequency: ω=π.
Dual: Fourier series: Expand X(ej ω) as a Fourier series with period 2π:
x(n)=Fourier coefficients; computed using DT F T −1formula above.
Uniform P|x(n)|<∞(absolutely summable) →unif orm convergence:
converge lim
N→∞ max |XN(ejω )−X(ejω )|= 0 where XN(ej ω) = PN
n=−Nx(n)ejωn
Mean- P|x(n)|2<∞(finite energy)→mean −square convergence:
square lim
N→∞ Rπ
−π|XN(ejω )−X(ejω )|2dω = 0. Weaker than uniform:
Sinc: x(n) = sin(Bn)
πn →X(ejω ) = 1 if 0 ≤ |ω|< B
0 if B < |ω| ≤ π
Finite x(n) = {...0,0,3,1,4,2,5,0,0...}(x(0) = 4; finite length = 5) →
length X(ejω ) = 3ej2ω+ 1ejω + 4 + 2e−j ω + 5e−j2ω=X(z)|z=ejω
signal X(ejω ) = [4 + 3 cos(ω) + 8 cos(2ω)] −j[sin(ω) + 2 sin(2ω)]
Expo- x(n) = anu(n)→X(ejω) = P∞
n=0 ane−jωn = 1/(1 −ae−jω).
nential x(n) = anu(n) + bnu(−n−1) →X(eω) = b−a
a+b−ejω −abe−jω
provided: |a|<1<|b|(stable x(n)⇔ROC must include the unit circle |z|= 1).
DISCRETE-TIME FOURIER SERIES (DTFS)
DTFS: Xk=1
NPN−1
n=0 x(n)e−j2πnk/N ;x(n) = PN−1
k=0 Xkej2πnk/N
Discrete+periodic in time domain⇔Discrete+periodic in frequency.
Basis: PN−1
n=0 ej2πnk/N =nNif N divides k
0 otherwise . Orthogonal function.
Periodic: x(n), Xk, ej2πnk/N are all periodic in nand kwith periods N.
Parseval: 1
NPN−1
n=0 |x(n)|2=PN−1
k=0 |Xk|2=power in the periodic x(n).
Square: x(n) = 1 if 0 ≤n≤L−1
0 if L≤n≤N−1→Xk=(L
Nif N divides k; else
1
N
sin(πkL/N)
sin(πk/N)e−jπk(L−1)/N
Continuous L ⇔ F
z=esms=jω m F {Px(n)δ(t−n)}
Discrete Z ⇔ DT F T
T ime z =ejω
