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SCEE08007 Signals and Communications 2
Formula and Tables of Transforms
Trigonometric Identities
sinx =ejx ejx
2jcosx =
ejx + ejx
2
1 = cos2 x+ sin2 x cos 2x = cos2 x sin2 x
cos2 x =1
2(1 + cos 2x) sin2 x =
1
2(1 cos 2x)
cos3 x =1
4(3 cosx+ cos 3x) tan (AB) =
tanA tanB
1 tanA tanB
cos(AB) = cosA cosB sinA sinB sin(AB) = sinA cosB cosA sinB
sinA+ sinB = 2 sin
(A+B
2
)cos
(AB
2
)sinA sinB = 2 cos
(A+B
2
)sin
(AB
2
)
cosA+ cosB = 2 cos
(A+B
2
)cos
(AB
2
)cosA cosB = 2 sin
(A+B
2
)sin
(AB
2
)
cosA cosB =1
2[cos(A+B) + cos(AB)] sinA cosB =
1
2[sin(A+B) + sin(AB)]
sinA sinB =1
2[cos(AB) cos(A+B)]
a cosx+ b sinx = c cos (x+ ) where c =a2 + b2 and = tan1
b
a
g
a
b
c
a
b
Cosine Rule: c2 = a2 + b2 2ab cos
Sine Rule:a
sin=
b
sin=
c
sin
Fourier Series Analysis of Periodic Waveforms
If g(t) is periodic with period T , then:
g(t) =a02
+
n=1
[an cos (n0t) + bn sin (n0t)]
where an =2
T
T2
T
2
g(t) cos (n0t) dt and bn =2
T
T2
T
2
g(t) sin (n0t) dt
or: g(t) =
n=
cn ejn0t where cn =
1
T
T2
T
2
g(t) ejn0t dt
where 0 =2piT
= 2f0; f0 =1
Tis the fundamental frequency.
Fourier Transform Analysis of Aperiodic Signals
The Fourier transform of a signal g(t) is given by:
G() =
g(t) ejt dt and g(t) =1
2
G() ejt d
Parsevals theorem of energy conservation:
|g(t)|2 dt =1
2
|G()|2 d
Selected Fourier Transforms
g(t) G()
1 (DC level) 2()
u(t) (unit step) () +1
j
ej0t 2( 0)
cos0t [( 0) + ( + 0)]
sin0t
j[( 0) ( + 0)]
n=
(t nT ) (impulse train)2
T
m=
(
2m
T
)
g(t ) (time shift) ej G()
g(at) (scale in time)1
|a|G(a
)
ej0t g(t) G( 0) (frequency shift)
g1(t) g2(t) (convolution) G1()G2() (multiplication)
g1(t) g2(t) (multiplication)1
2G1() G2() (convolution)
Duality: If g(t) transforms to p(), then p(t) transforms to 2g().
Symmetry: If g(t) is real, then G() = G() ( means complex conjugate).
If g(t) is real and even, then G() is real and even.
If g(t) is real and odd, then G() is imaginary and odd.
z-Transforms
The z-transform of a discrete-time causal sequence
g[n] (defined for n = 0, 1, 2, . . . ) is given by:
G(z) =
n=0
g[n] zn
Selected z-Transforms
g[n], (n 0) G(z)
[n] (unit pulse) 1
[nm] zm
1 (unit step)z
z 1
n (unit ramp)z
(z 1)2
rnz
z r
n rnr z
(z r)2
sin(0n)z sin0
z2 2z cos0 + 1
cos(0n)z2 z cos0
z2 2z cos0 + 1
rn sin0nz r sin0
z2 2z r cos0 + r2
rn cos0nz2 z r cos0
z2 2z r cos0 + r2
rn g[n] G(r1z
)
g[n+ 1] z G(z) zg(0)
g[n 1] z1G(z) + g(1)
Final Value Theorem:
limn
g[n] = limz1
(z1)G(z) (discrete-time)
Communications Theory
Amplitude Modulation An amplitude modu-
lated (AM) signal can be expressed as:
xc(t) = Ac [1 + amn(t)] cos (2fct)
Angle Modulation The general angle-modulated
signal is given by:
xc(t) = Ac cos (2fct+ (t))
For PM: (t) = kpm(t)
For FM: (t) = kp
tm() d
= 2fd
tm() d
Angle Modulation
(t) = sin (2fmt)
the angle-modulated signal is:
xc(t) = Ac
n=
Jn () cos [2 (fc + nfm) t]
where Jn () =
Jn () , for n even
Jn () for n odd
The bandwidth is given by Carsons Rule:
B 2 ( + 1) fm
For a FM modulator withm(t) = A cos (2fmt),
= fdA/fm
Carsons Rule For Arbitrary FM Signals:
B 2 (D + 1)W
where W is the bandwidth of the message signal
m(t), and the deviation ratio is
D =peak frequency deviation
W
C = B log2 (1 + S/N) bit/s Q = wr/BW = wrL/R = wrCR