Digital Signal Processing Reference
In-Depth Information
Two important questions arise at this time:
•
What is the appropriate value of the sampling interval,
T
sec., or
inversely, what is the appropriate value of the sampling frequency
f
s
= 1/
T
in cycles per sec. or Hertz (Hz)?
•
Is it possible to recover
x
(
t
) exactly from the sample values
x
(
n
):
n
=
0, 1, 2 …
N
- 1?
The answer to the first question is given by the Nyquist Sampling Theorem,
which states: If
x
(
t
) is a bandlimited signal with the maximum signal fre-
quency
Ω
m
, then
x
(
t
) is uniquely determined from its samples
x
(
n
):
n
= 0, 1,
2 …
N
- 1, if the sampling interval
T
≤
π
/
Ω
seconds, or, alternately, if the
m
sampling frequency
f
s
Hz.
The answer to the second question is given by the interpolation formula
given below in Equation 4.1. If the sampling satisfies the Nyquist sampling
theorem, then the recovered signal values (between the samples) is given by:
≥
Ω
/
π
m
−
∑
N
1
( )
sin
⎣
π
tnTT
tnTT
(
−
)
⎦
xt
()
=
xn
(4.1)
r
π
(
−
)
n
=
0
which will be studied in Section 4.2, is different
than the ideal sampling described in this section and by Equation 4.1. One
of the practical problems in ideal sampling is the impossibility of generating
ideal impulses with zero time width.
However,
practical sampling,
4.1.1.2
Amplitude Quantization
The second stage in the A/D process is amplitude quantization, where the
sampled discrete-time signal
x
(
n
)
, n=
0, 1, 2
… N
- 1 is quantized into a finite
set of output levels
ˆ
()
,
n =
0, 1, 2 …
N
- 1. The quantized signal
ˆ
()
can
xn
xn
take only one of
L
levels, which are designed to cover the dynamic
range
is the maximum amplitude of the signal.
Both uniform and nonuniform quantizers will be considered in this section.
, where
x
M
−≤
xxnx
M
()
≤
M
Uniform Quantizer
The design of an
L-level uniform quantizer
is detailed below in a 4-step process.
Step 1: Dynamic range of the signal
Fix the dynamic range of the sampled signal -
x
M
≤
x
(
n
)
≤
x
M
.
Step 2: Step size of quantizer
The step size of the uniform quantizer is given as:
