IEEE Signal Processing - May 2018 - 21
Scalar Quantization
Consider the problem of representing a random number X drawn from a continuous distribution using another number taken from a finite alphabet of K elements
X Q ! " x 1, f, x K ,. Since an exact representation of X cannot be attained due to cardinality limitations, the goal is
to minimize
E ^X - X Q h2.
(S1)
The mapping of X to XQ is called quantization. When the
representation of a sequence of random numbers is considered, we use the term scalar quantization to denote that
the same quantization mapping is applied to each element
of the sequence, independently of the previous elements.
Assuming that the quantizer inputs are independent, the
estimation of each input sample from the output of the quantizer is based on only one of these K states " x 1, f, x K ,.
Evidently, the minimal estimation error is attained by mapping X to the reconstruction value xi that minimizes (S1). As
a result, the procedure of optimizing a scalar quantizer of K
states can be described by selecting the optimal K reconstruction values. Given the distribution of the input, this optimal set may be attained by an iterative procedure known
as the Lloyd algorithm or, more commonly, the K-means
algorithm [30], [31].
The number of bits or the bit resolution of the quantizer
is the number of binary digits that represent X at its output by assigning a different label to each state. Clearly,
the output of a K-state quantizer can be encoded with
^log 2 K h binary digits. However, this number may be
reduced on average if the labels of the states consist of
binary numbers of different length. For example, by using
and only in the case where fs is smaller than the Nyquist rate
of X (t). In fact, this distortion is exactly the energy in the part
of the spectrum of X (t) blocked by the presampling filter. We
therefore have
fs
2
fs
2
D smp ( fs) _ v 2 - #
S X ( f ) df.
Note that D smp ( fs) equals zero when fs is above the Nyquist
rate of X (t).
To analyze the distortion due to quantization, we represent
the output of the quantizer as
YQ [n] = Y [n] + h [n], n = 0, 1, f,
(2)
where h [n] = YQ [n] - Y [n] is the quantization noise. Since
there is no aliasing in the sampling operation, the reconstruction filter applied to Y [n] leads to the signal X p (t) at the out-
uniform quantization levels to quantize a nonuniformly
distributed input, we may label those states that are more
likely with binary numbers shorter than those numbers
assigned to less likely states. These numbers must satisfy
the condition that no member is a prefix of another member, so that the sequence of states can be uniquely
decoded. This procedure is denoted as variable-length
scalar quantization, distinguished from fixed-length quantization, in which the labels are all binary numbers of the
same length.
Interestingly, the average mean squared error (MSE) over
an independent and identically distributed sequence using
a variable-length scalar quantizer may be strictly smaller
than with a fixed-length scalar quantizer for the same average number of bits, even if the levels in the latter were
optimized for the input distribution using the Lloyd algorithm. For example, with input taken from a standard
normal distribution, the average MSE attained by a variable-length scalar quantizer with equally spaced reconstruction levels and an optimal labeling of these levels
r
r
converges to (re/6) 2 -2R . 1.42 # 2 -2R as Rr becomes
large [32]. In fact, it is also shown in [32] that a uniform
quantizer with optimal labeling converges to the optimal
variable-length quantizer as Rr increases. However, the distortion attained by a fixed-length quantizer under an optir
mal selection of the K = 2 R reconstruction levels converges
r
-2Rr
to ( 3 r/2) 2 . 2.72 # 2 -2R [30].
As explained in "Source Coding and the DRF," a lower
MSE for the same average number of bits per source sample Rr can be attained by using a vector quantizer, i.e., by
considering the joint encoding of multiple samples from a
sequence of samples, rather than one sample at a time.
put of the first LPF. Since the quantizer is a deterministic
function of Y [n], the process h [n] is stationary, and we
denote its PSD by S h ( f ) [note that S h ( f ) is periodic, with a
period of fs ]. Nevertheless, an exact description of the statistics of h [n] turns out to be a surprisingly difficult task. As a
result, many approximations of its statistics have been developed [17], [19]. Most of these approximations provide conditions under which the spectrum of h [n] is white (i.e.,
different elements of h [n] are uncorrelated) [20]. One of the
widely used approximations was provided by Bennet [21],
who showed that, when the distribution of the input to the
quantizer Y [n] is continuous and the quantization levels are
uniformly distributed, the spectrum of the quantization noise
S h ( f ) converges to a constant as the quantizer resolution Rr
increases. Another way to achieve the uniform spectral distribution of h [n] is by dithering the signal at the input to the
quantizer, i.e., by adding a pseudorandom noise signal [22].
For simplicity, our following analysis assumes that S h ( f ) is a
IEEE Signal Processing Magazine
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May 2018
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21
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