IEEE Solid-States Circuits Magazine - Spring 2018 - 11

A C irCu it for All SeA SonS

Behzad Razavi

The Biquadratic Filter

T

The biquadratic filter, also known as
the "biquad," dates back to the 1960s
[1]-[3] but still serves as an essential
building block in analog filter design.
In this article, we study this circuit's
properties and design issues.

H (s) =

=

The General Biquad
The biquad is a second-order filter
whose transfer function is given, in
the general case, by
2
H (s) = a 1 s 2 + b 1 s + c 1 .
a2 s + b2 s + c2

characterized by the following two
(equivalent) transfer functions:

(1)

Here, the numerator coefficients can
be chosen to yield a low-pass, bandpass, or high-pass response. For example, a 1 = b 1 = 0 leads to a low-pass
filter (LPF), the focus of our study
here. To realize higher-order filters,
biquad sections can be cascaded.

The Need for Complex Poles
We typically begin the design of filters
by deciding on the order and shape of
their frequency response. For example,
Wi-Fi receivers commonly employ a
fifth-order LPF to suppress unwanted
channels. However, for a given order,
the roll-off from the passband to the
stopband can be made sharper if some
peaking or ripple is tolerable.
It is in this spirit that we turn to
transfer functions having complex
poles. We explain the thought process
behind this point by means of an example. Consider a low-pass biquad

~ n2
~n
2
s +
s + ~ 2n
Q

~ p1, 2 = -

(4)

taking on a complex value if Q > 1 2 .
But complex poles do not necessarily imply peaking. Writing

(2)

~ 2n
.
s 2 + 2g~ n s + ~ n2

1- 12,
4Q

~n
! j~ n
2Q

(3)

H ( j~)

Here, ~ n denotes the natural frequency, Q the quality factor (also
called the pole Q ), and g the damping
factor. The first form is common in filter design and the second in control
theory (e.g., in phase-locked loops).
Noting that Q = 1/ (2g), we will use
the two forms interchangeably.
We intuitively observe that, if Q " 3,
the two poles approach ! j~n and the
system becomes unstable. Thus, the
value of Q determines how much the
poles depart from the real axis and how
much peaking H (s = j~) has. The two
poles can be expressed as

2

=

~ n4
^~ 2n

2

- ~ h + c ~n ~ m
Q
2 2

, (5)

we find that the denominator falls to
a minimum at ~ a = ~ n 1 - 1/ ^2Q 2h
if Q 2 2 /2. In such a case, H ( j~)
exhibits a peak equal to Q
1 - 1/ ^4Q 2h (Figure 1). It is helpful
to remember that for Q = 1, the peaking is about 1.15 dB, and it occurs at
~ a = 0.71~ n .
With these preliminary developments, we ca n now eva luate the
stopband attenuation of H at a
given frequency for the case of real
or complex poles. As an example,
we seek the rejection at a frequency

H (ω )
Q
1−

1
4Q 2
Q>

1
Q<

2

2
2

2

ωa = ωn

1−

1
2Q2

ω

Digital Object Identifier 10.1109/MSSC.2018.2822859
Date of publication: 22 June 2018

Figure 1: Peaking in biquad frequency response.

IEEE SOLID-STATE CIRCUITS MAGAZINE

s p r i n g 2 0 18

11
Table of Contents for the Digital Edition of IEEE Solid-States Circuits Magazine - Spring 2018
