fundamentals of engineering supplied-reference handbook - Ventech!

Therefore, the impulse response h(t) and frequency response
H(f) form a Fourier transform pair:
Parseval’s Theorem
h( t)
↔ H ( f )
The total energy in an energy signal (finite energy) x(t) is
given by
+∞ 2 +∞ 2
∫ ∫
E = x() t dt = X( f) df
∫
−∞ −∞
+∞
= G ( f) df =φ (0)
−∞
xx xx
Parseval’s Theorem for Fourier Series
As described in the following section, a periodic signal x(t)
with period T0 and fundamental frequency f0 = 1/T0 = ω0/2π
can be represented by a complex-exponential Fourier series
n=+∞
xt () = ∑ Xne n=−∞
jn2πf0t The average power in the dc component and the first N
harmonics is
n=+ N
∑
2
n
2
0
n= N
2 ∑
2
n
n=− N n=
0
P = X = X + X
The total average power in the periodic signal x(t) is given
by Parseval’s theorem:
1 t0+ T
n=+∞
0 2
= ()
T ∫
=
t ∑
0 0
n=−∞
P x t dt X
AM (Amplitude Modulation)
x () t = A [ A+ m()]cos(2 t πf
t)
AM c c
'
c n c
= A [1 + am ( t)]cos(2 πft)
2
n
The modulation index is a, and the normalized message is
mt ()
mn() t =
max mt ( )
The efficiency η is the percent of the total transmitted power
that contains the message.
2 2
n
2 2
n
a < m () t >
η= 100 percent
1 + a < m ( t)
>
where the mean-squared value or normalized average power
in mn(t) is
2 1 + T 2
< mn() t >= lim mn() t dt
T →∞ 2T
∫ −T
If M(f) = 0 for |f | > W, then the bandwidth of xAM(t) is 2W.
AM signals can be demodulated with an envelope detector
or a synchronous demodulator.
177
ELECTRICAL AND COMPUTER ENGINEERING (continued)
DSB (Double-Sideband Modulation)
x () t = A m()cos(2 t π f t)
DSB c c
If M(f) = 0 for |f | > W, then the bandwidth of mt () is W and
the bandwidth of xDSB(t) is 2W. DSB signals must be
demodulated with a synchronous demodulator. A Costas
loop is often used.
SSB (Single-Sideband Modulation)
Lower Sideband:
Upper Sideband:
⎛ f ⎞
xLSB () t ↔ XLSB ( f) = XDSB( f)
Π⎜ ⎝2f⎟ ⎠
⎡ ⎛ f ⎞⎤
xUSB ( t) ↔ XUSB ( f) = XDSB( f)
⎢1−Π⎜ ⎥
2 f ⎟
⎣ ⎝ c ⎠⎦
In either case, if M(f) = 0 for |f | > W, then the bandwidth of
xLSB(t) or of xUSB(t) is W. SSB signals can be demodulated
with a synchronous demodulator or by carrier reinsertion
and envelope detection.
Angle Modulation
x () t = A cos[2 π f t +φ ()] t
Ang c c
The phase deviation φ(t) is a function of the message m(t).
The instantaneous phase is
φ () t = 2 π f t+φ () t radians
i c
The instantaneous frequency is
d d
ω i() t = φ i() t = 2 π fc + φ () t radians/s
dt dt
The frequency deviation is
d
∆ω () t = φ () t radians/s
dt
PM (Phase Modulation)
The phase deviation is
φ () t =
k m() t radians
P
c