§1 / 1 OUTLINE
A filter affects different frequencies of a signal in different ways, by attenuation and phase shift. It pretty much makes sense to talk about filters only in terms of linear circuits. (What can happen to frequencies after they pass through a nonlinear operator?) The main use of a filter is to reduce "noise" in the signal due to power at unwanted frequencies (60 cycle noise, for example).
Why filter? need for anti aliasing
analog vs digital filters. Filters in LabVIEW:
the problem of inductors
passive vs active filters
optimal vs intuitive (RC) filters
first order filters: LP and HP
bandpass: LP and HP in series
Q: band-reject: LP and HP in parallel, converging on summation
Use of superposition
Distorted step response due to phase effects of filter
optimal higher order filters: Butterworth, Chebyshev, Bessel,
Gaussian
Handouts on active filters. Chapt 5 H & H
Recall Circuits information about Bode plots, and how to sketch gain and phase of a transfer
function.
All LP filters will roll off at -20dB/decade per pole, when well above the cutoff frequency(ies).
Or at 6 dB per octave.
Can use Matlab to generate exact spectrum
§1 / 2 Reading
Horowitz & Hill, The Art of Electronics (2nd Ed), Cambridge Univ Press (1989). Chapter 5, "Active Filters and Oscillators."
From EN157: Oppenheim & Willsky, Signals and Systems, chapter 6, "Filtering" (1983)
Poularikas & Seely (URI), Signals and Systems, chapter 11, "Digital Filtering" (1985)
See p. 568, Chebyshev filter form.
§1 / 3 Filter specifications
Magnitude of response: Frequency range from passband, to stopband.
The spec will say what attenuation is required for what frequencies. Outside the
attenuations, it is expected that the gain will be 1, no attenuation. The falloff of the filter will
determine the Order of the filter. (How many capacitors or inductors (energy storage
elements) it has, number of poles in transfer function.) Sometimes the spec will require
flatness in the passband.
Should the filter be adjustable? If so, there may be designs which allow one potentiometer to be changed, that changes the frequency range. Filter should also be insensitive to changes in components due to heating or aging.
Phase characteristics: If different frequencies are delayed different times then the output of the filter will not be a faithful version of the input. Most clearly seen in response to step input, where overshoot and ringing may occur after filtering. Why does linear phase relationship mean that delay is constant? Suppose delay through the filter is 10 microseconds; 10 microseconds is a linearly increasing percentage of phase of increasing frequency. Thus linear phase is the goal, and Bessel filter is best for that.
§1 / 4 Filter types
Ideal filter: step like transitions: Impossible, but...
Various filters will help you meet different specifications.
Filter synthesis: Wonderful topic from years gone by in EE curriculum: place frequencies in a
Laplace transform Zeros(s)/Poles(s) transfer function.
For an active filter, consider Z(s) as the the feedback circuit, and P(s) as the source circuit in a
negative gain summation amp.
From EN52, you learned to make Bode plots of such transfer functions.
Here in 123, we will consider well-known analog filter types, and learn to use tables and
templates to meet filter specs.
§1 / 4 / 1 Butterworth filter
It's a maximally flat filter. In the passband the gain is optimally constant. The phase properties
are not ideal, like Bessel, and the sharpness of cutoff is not as good as Chebychev. The
Butterworth gain formula is
§1 / 4 / 2 Chebyshev filter
The gain formula is 
Notice the normalizing f/fc term is not taken to a power!
and that the way to "unnormalize" the polynomial Cn is divide frequency by fc.
Epsilon controls the amount of ripple in the passband. See Poularikis & Seely, p. 569, and
program filterA.m on the 7100 Mac. Is the frequency in rad/sec or in Hz? In the
development in H&H page 275, the formulas are given in Hz, so 2p is needed in the
conversion to R and C in the circuit.
Chebyshev polynomials:
for n = 2 
for n = 4 
for n = 6 
§1 / 4 / 3 Bessel filter and good time domain response.
A Bessel filter has maximally flat time delay, results in linear phase. As frequency increases, a constant delay becomes an increasingly greater fraction of 360 degrees. If delay at different passband frequencies is not constant, then the resulting waveform will be distorted. Figure 5.14 in H&H, comparing Butterworth vs Bessel. See step response for example, Fig. 5.15.
VCVS circuit as a template for Bessel. Can use Table 5.2 for Bessel design.
The Gaussian filter: See Discussion in H&H, p. 272. Another filter with good phase characteristics.
§1 / 5 Designing filters
§1 / 5 / 1 Avoiding inductors
Inductors: RLC filters can provide excellent characteristics, and they are the analytical subject of RLC analysis in a Circuits class. BUT inductors are not structurally "flat", and have "parasitic" capacitance and conductance. Their real properties can degrade the performance of real filters. From H&H p. 266, on inductors: "They are often bulky and expensive, and they depart from the ideal by being 'lossy', .i.e., by having significant series resistance and other 'pathologies' such as nonlinearity, distributed winding capacitance, and susceptibility to magnetic pickup of interference." Goal: make good filters without the need for real inductors.
One solution: Create op amp circuits (active filters) that mimic inductors: Start with a NIC:
Negative Impedance Converter. Figure 5.4 from H&H is wrong: the top resistor should be Z
and Z should be R. Compare to Roberge book, page 457.
Current flowing through the bottom Z, due to Vout, must flow out the Vin line...
See Roberge, p. 455-56 for development. the gain is +2 and the current flow from Vout to Vin
is Vin/Z. Iin is then -Vin/Z so Zin = -Z.
Roberge on pp 456-57 analyzes the gryrator, that creates the inverse of an impedance.
End up with the equation 
§1 / 5 / 2 Q (Quality factor) of an inductor or a filter
Q defines the sharpness of a bandpass filter, and is a term introduced in study of second order
filters. Let b be the bandwidth of the filter and w the center frequency: 
p. 516 of H&H. Three forms of VCVS: Where is design for bandpass?
§1 / 7 First order filters
Radians/sec vs Hertz. The w in transfer functions is normally radians per second. You will be more interested in Hz (cycles/sec), so must use a 2p conversion at some point in your filter design process.
An active LP filter.
§1 / 7 / 1 Example of a HP Bessel filter design
Suppose you want to filter frequencies below 60 Hz. You are given the criteria that at 0.5x of the cutoff (30 Hz) the response is to be down by 10dB (0.3 attenuation). You are told to design a filter with good delay: linear phase. For construction, you have a variety of resistors, but only 0.3 mF capacitors to work with.
You decide to "design" a VCVS filter, using the information in Table 5.2 of H&H. You select a
Bessel filter for good delay characteristics. Look on the graphs of page 276 H&H and see that
at 2 x 60 Hz a 4th order filter is needed; secon order doesn't have enough attenuation at 0.5x.
You must use the 2x reading, since the graphs are for LP forms.
On page 275 H&H say, "To make a high-pass filter, use the high-pass configuration shown
previously, with the [front end] Rs and Cs interchanged." See Fig 5.16 for circuit. For the Bessel
filters the normalizing factor fn must be inverted.
In table 5.2 you see that for 4 poles the Bessel terms are
K=1.084 and 1.759, and the fn terms are 1.432 and 1.606 respectively.
Invert the fn for 0.698 and 0.623.
For the first stage, if R = 10K, then (K-1)*R = 840 ohms
For the second stage, if R = 10K, then (K-1)*R = 7.6K ohms
For each stage, want R1 = R2 and C1 = C2 (Fig 5.16)
Solve for 
= .0038; If we say C = 0.3 x 10-6, then R1 = 12.666K
Now solve for 
= .0043; If we say C = 0.3 x 10-6, then R2 = 14.12K
Done!
§1 / 8 Bode plots
For specifying the filter characteristics of amplitude and phase vs freq.
See page 704 of Nilsson (4th)
A log-log plot
Real poles and zeros; complex poles and zeros.
First order plot: at w0=w the T(s) is -3dB, at 10x a pole is down -20dB, at 100x down -40dB,
not a sharp skirt...
Use of Matlab.
§1 / 8 / 1 Bandpass and notch filters
Combine HP and LP filters
two filters, LP, BP in series for bandpass,
band reject filter: two filters in parallel, summed for output.
Always a bandpass: For the circuit below, say Cs are fixed, maybe C1 = C2 . Now say
R1 = 10K and R2 = 1K, Question what kind of filter is represented? You'd think, from
computing the HP for Zs and the LP for Zf that it would be a band reject filter, but if you
plot out gain vs freq with a program like Matlab (en123PS398.m) you'll see it is a bandpass
filter.
The strange thing is, if you reverse the resistors, and now make R1 = 1K and R2 = 10K, you
get exactly the same gain vs freq curve! Again a bandpass. But if you put the parallel
combination where Zs goes, and the series combination where Zf goes, you can come up
with a band-reject filter.
Quality factor Q of a filter is the ratio of bandwidth to resonant frequency, of an RLC circuit.
need more. See EN 52 book.
Summary List
* Filters for anti-aliasing
* Analog vs digital filters
* Active filters with op amps
eliminate the need for inductors
* Specifications for filters
basic RC for first order LP and HP filters
gain, phase
* Making bandpass and band-reject filters from LP and HP blocks.
* Use of standard VCVS designs to meet specs.