Real-time digital signal processing: implementations, ... changes in the input signal is limited by its internal clock rate, so that it may be slow to
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A.4 Impulse Functions
The unit impulse function dt can be defined as
dt 1, if t 0
0, if t T 0.
A:26
Thus we have
I
ÀI
dtdt 1
A:27
and
I
ÀI
dt À t
0
xtdt xt
0
,
A:28
where t
0
is a real number.
A.5 Vector Concepts
Vectors and matrices are often used in signal analysis to represent the state of a system
at a particular time, a set of signal values, and a set of linear equations. The vector
concepts can be applied to effectively describe a DSP algorithm. For example, define an
LÂ1 coefficient vector as a column vector
b b
0
b
1
. . . b
LÀ1
T
,
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where T denotes the transpose operator and the bold lower case character is used to
denote a vector. We further define an input signal vector at time n as
xn xn xn À 1 . . . xn À L 1
T
:
A:30
The output signal of FIR filter defined in (3.1.16) can be expressed in vector form as
yn
X
LÀ1
l0
b
l
xn À l b
T
xn x
T
nb:
A:31
Therefore, the linear convolution of an FIR filter can be described as the inner (or dot)
product of the coefficient and signal vectors, and the result is a scalar y(n).
If we further define the coefficient vector
a a
1
a
2
Á Á Á a
M
T
A:32
VECTOR CONCEPTS
449
Summary :
xn À L 1 T : A:30 The output signal of FIR filter defined in (3.1.16) can be expressed in vector form as yn X LÀ1 l0 b l xn À l b T xn x T nb: A:31 Therefore, the linear convolution of an FIR filter can be described as the inner (or dot) product of the coefficient and signal vectors, and the result is a scalar y(n).
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ector,signal,concepts,define,coefficient,ectors,linear,impulse,further,defined,là1,filter,used