IEEE Circuits and Systems Magazine - Q1 2018 - 11
I. Introduction
compressing and communicating data. It also defines the
IR filters are commonly used in a wide range of sig- notion of channel capacity and provides a mathematical
nal processing applications including wireless and model by which one can compute it. Channel capacity is
wireline communications, biomedical systems and the tightest upper bound on the rate of reliable informadata converters [1]-[10]. Therefore minimizing the hard- tion transmission over a communication channel. A key
ware complexity of FIR filters is of considerable interest in measure of information is entropy, which is usually exmany applications. In this paper we investigate the follow- pressed by the average number of bits needed to store or
ing general questions:
communicate one symbol in a message [14]-[17]. The ap1) Given a fixed hardware (HW) budget (i.e., a maxi- plication of the channel capacity concept to an additive
mum allocated HW complexity) can we define a white Gaussian noise (AWGN) channel, assuming a bandpractical upper bound for the achievable Parks- width of B (Hz), an average received signal power P
McClellan-Remez order [11]-[13] of a direct form (Watts) and the noise power spectral density N 0 (W/Hz)
results in the following upper bound on the rate of inforFIR filter?
2) If yes, how close to such a practical limit are the mation in the AWGN channel as shown in Fig. 1 (known as
the Shannon-Hartley theorem [14]-[16]):
hardware complexities of existing filter designs?
We believe that such intuition would be valuable
Rate of information (bits/ sec) # B log 2 c 1 + P m (1)
because, given a target FIR filter specification (passN0 B
band ripple, passband edge, stopband attenuation
and stopband edge), it would provide a guideline where the second term in parenthesis P/N 0 B is the re(measure of goodness) for the required lowest design ceived signal-to-noise power ratio (SNR). In summary,
complexity level, which may also hint at the best- the available SNR (the budget) at the receiver side decase (lowest possible) power consumption given a fines the upper bound for the rate of information that
specific technology node that is chosen to implement can be communicated in the channel.
With the above insights in mind, we present a similar
the filter. We believe this to be an important practical
guideline in order to understand and properly budget approach for the implementation of FIR filters, where we
the power and the design complexity for a required can envision that the available hardware budget (HW
FIR filter, given the overall system design constraints complexity) defines an upper bound for the Remez or(e.g., for a high-order sharp transition-band filter in a der of the practically realizable FIR filter (Fig. 1). One
can also think of this requirement as the minimal hardlow-power system).
Before delving into FIR filter details, and in order to ware resources that are required to enforce the desired
further highlight the importance of such insight, it is frequency-domain relationship (as defined by the given
perhaps worthwhile to observe the
si milarity between this concept
and Shannon's "channel capacity"
16
[14]-[15] which, since 1948, has
Impractical
Region
8 Impractical
served as the golden standard for
Region
4
the upper bound of the achievable
?
data rate in communication chan2
?
nels. We illustrate this similarity in
Practical
1
?
Region
Fig. 1. The landmark event that es1/2
tablished the discipline of informa1/4
tion theory [14]-[17], and brought
1/8
it to immediate worldwide attention, was the 1948 publication of
0
10
20
30
Shannon's classic paper "A MathPower Budget: SNR (Eb/N0)(dB)
Hardware Budget
ematical Theory of CommunicaFigure 1. the available power budget defines the maximum realizable data rate in
tion" in The Bell System Technical
AWgN channel (shannon's channel capacity). given a hardware budget can we also
Journal [14]. Information theory dedefine a practical upper bound for the realizable fIr filter?
scribes the fundamental limits on
Maximum Realizable FIR Filter Order
Shannon-Hartley
Channel Capacity/Hz
Realizable Data Rate/Hz (bits/s/Hz)
f
Alireza Mehrnia is with the Department of Electrical Engineering, University of California, Los Angeles, CA 90095 USA (e-mail: mehrnia@ucla.edu).
Alan N. Willson, Jr. is the Charles P. Reames Professor Emeritus at the Department of Electrical Engineering, University of California, Los Angeles, CA
90095 USA (e-mail: willson@ee.ucla.edu).
fIrst quArtEr 2018
IEEE cIrcuIts ANd systEMs MAgAzINE
11
Table of Contents for the Digital Edition of IEEE Circuits and Systems Magazine - Q1 2018
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