fundamentals of engineering supplied-reference handbook - Ventech!

DIGITAL SIGNAL PROCESSING
A discrete-time, linear, time-invariant (DTLTI) system with
a single input x[n] and a single output y[n] can be described
by a linear difference equation with constant coefficients of
the form
y
k
l
[ n]
∑by[ n − i]
= ∑ax[ n − i]
+ i
i
i=
1 i=
0
If all initial conditions are zero, taking a z-transform yields a
transfer function
H
( z)
Y
=
X
∑ a z
i
( z)
i=
0
=
k ( z)
k
z + ∑
l
k −i
k −i
bi
z
i=
1
Two common discrete inputs are the unit-step function u[n]
and the unit impulse function δ[n], where
u
⎧0
⎨
⎩1
n < 0⎫
n ≥ 0
⎬
⎭
[] n =
and δ[]
n
⎧1
= ⎨
⎩0
n = 0 ⎫
n ≠ 0
⎬
⎭
The impulse response h[n] is the response of a discrete-time
system to x[n] = δ[n].
A finite impulse response (FIR) filter is one in which the
impulse response h[n] is limited to a finite number of points:
h
k
[] n ∑ a δ[
n − i]
= i
i=
0
The corresponding transfer function is given by
H () z
k
∑ a
−i
= i z
i=
0
where k is the order of the filter.
An infinite impulse response (IIR) filter is one in which the
impulse response h[n] has an infinite number of points:
h
[] n a δ[
n − i]
∑ ∞
= i
i=
0
175
ELECTRICAL AND COMPUTER ENGINEERING (continued)
COMMUNICATION THEORY AND CONCEPTS
The following concepts and definitions are useful for
communications systems analysis.
Functions
Unit step,
u (t)
Rectangular
pulse,
Π(/ t τ )
Triangular pulse,
Λ(/ t τ )
Sinc,
sinc( at )
Unit impulse,
δ () t
The Convolution Integral
In particular,
∫
+∞
⎧⎪
0 t < 0
ut () = ⎨
⎪⎩ 1 t > 0
⎧
1
⎪1
t / τ <
⎪
2
Π(/ t τ ) = ⎨
⎪
1
⎪⎩
0 t / τ >
2
⎧
⎪
1 − t/ τ t/
τ < 1
Λ(/ t τ ) = ⎨
⎪⎩ 0 t / τ > 1
∫
sin( aπt) sinc( at)
=
aπt +∞
−∞
x( t+ t ) δ ( t) dt = x( t )
0 0
for every x(t) defined and
continuous at t = t0. This is
equivalent to
∫
+∞
−∞
xt () ∗ ht () = x( λ) ht ( −λ) dλ
−∞
∫
x() t δ( t − t ) dt = x( t )
+∞
0 0
= ht ( ) ∗ xt ( ) = h( λ) xt ( −λ) dλ
−∞
x() t ∗δ( t − t ) = x( t− t )
0 0
The Fourier Transform and its Inverse
+∞ − j2πft −∞
X( f) = x( t) e dt
∫
∫
+∞ j2πft −∞
x() t = X( f) e df
We say that x(t) and X(f) form a Fourier transform pair:
x() t ↔
X( f)