563489578934

TABLE 2–2
SOME FOURIER TRANSFORM PAIRS
Function Time Waveform w(t) Spectrum W( f )
Rectangular ∑ a t T b T[Sa(pfT)]
Triangular a t T b T[Sa(pfT)]2
Unit step u(t) ! e + 1, t 7 0
0, t 6 0
Signum sgn(t) ! e + 1, t 7 0 1
-1, t 6 0 jpf
Constant 1 d (f)
1
2 d(f) + 1
j2pf
Impulse at t = t 0 d(t - t 0 ) e -j2pft 0
Sinc
Phasor
Sa(2pWt)
e j(v 0t+w)
1
2W ∑ a f
2W b
e jw d(f - f 0 )
Sinusoid cos (v c t + w)
1
2 ejw d(f - f c ) + 1 2 e-jw d(f + f c )
Gaussian e -p(t>t 0) 2 t 0 e -p(ft 0) 2
Exponential, one-sided e e-r>T , t 7 0
0, t 6 0
T
1 + j2pfT
Exponential, two-sided
-|t|>T
e 2T
1 + (2pfT) 2
Impulse train
k= q
a
k=-q
d(t - kT)
n= q
f 0 a
n=-q
d(f - nf 0 ),
where f 0 1/T
1.0
0.5
10 1
1
2pz
e z2 /2
10 2
Q (z)
10 3
10 4
10 5
10 6
10 7
Figure B–7
10 8
0
1 2 3 4 5 6
The function Q(z) and an overbound,
z
1
12p z e-z2 >2 .