FIR Filters - University of MemphisFIR →Finite Impulse Response Finite duration time-domain impulse response, often ten to a few hundred points, applied with direct time-domain convolution.

FIR Filters

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Mitch Withers, Res. Assoc. Prof., Univ. of Memphis

See Aster and Borchers, Time Series Analysis, chapter 5.

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Ut quod ali cibus est aliis fuat acre venenum.What is food to one, is to others bitter poison.Titus Lucretius Carus (c. 99 BC – 55 BC), Roman poet and philosopher.

An alternative interpretation: One person’s noise is another person’s signal.

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Digital Filtering is a very broad topic some spend entire careers on. We’ll be taking a cursory glance over a few short weeks.

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Filtering provides a means to alter the amplitude and/or phase of different frequency components of the data.

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1. Noise rejection/Signal Enhancement


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2. Remove instrument response from data


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3. Differentiate or integrate


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4. Change sampling rate


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4. Change sampling rate

5. Creative effects in audio and video


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One way to describe some filters we work with most frequently is what they do to the amplitude

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𝑓

Φ(𝑓)

A low pass filter keeps low frequencies and rejects high frequencies.

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𝑓

Φ(𝑓)

A low pass filter keeps low frequencies and rejects high frequencies.

Beware sharp edges (distortion)!

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𝑓

Φ(𝑓)

A high pass filter rejects low frequencies and keeps high frequencies.


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A band pass filter keeps frequencies within a specified range and rejects those outside the range.

𝑓

Φ(𝑓)


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𝑓

Φ(𝑓)

A notch or band gap filter rejects frequencies within a specified range and keeps those outside the range.


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The key in the design is enhancing the desired portion of the signal while minimizing distortion.

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Most filters we encounter are also Linear Time Invariant systems (LTI).

𝜙X Y

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𝜙X Y

Allows us to take advantage of all the tools we’ve learned so far.

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𝜙X Y

Allows us to take advantage of all the tools we’ve learned so far.

Beware the unstable filter (no poles on the RHS of the S-plane).

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One of the most simple, though not the best, ways to filter is by direct manipulation in the frequency domain.

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Computing the FT may be impractical for very long time series.

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Not real-time.

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Not real-time.

Windowing cautions apply.

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Not real-time.

Windowing cautions apply.

Run matlab command, filterdemo_mac

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FIR → Finite Impulse Response

Finite duration time-domain impulse response, often ten to a few hundred points, applied with direct time-domain convolution.

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Fast → no FT required.

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Real-time → Can be done in real-time using a small buffer (which means it can be acausal if you’re not careful).

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Stable → no recursive components. Time domain impulse response goes to 0 within finite number of points.

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Stable → no recursive components. Time domain impulse response goes to 0 within finite number of points.

Linear phase response → minimal distortion

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In the construction of a FIR filter, we typically select some desired frequency response, Ω(𝑓), and a desired number of points, 𝑀.

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Choice of 𝑀 is key and can either be done by truncation (aka, windowing with a boxcar), or by windowing in the time domain to 𝑀-points.

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Alternatively, choose the desired frequency-domain resolution, ∆𝑓, to determine 𝑀.

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Alternatively, choose the desired frequency-domain resolution, ∆𝑓, to determine 𝑀.

Both approaches accomplish the same thing, choosing 𝑀, but depend on which aspect is most important to the design; time-length or frequency resolution.

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Consider the running mean filter, possibly the most common FIR filter.

𝑤! =1𝑀Π" =

1𝑀 , 𝑛 ≤ 0𝑀 − 1

2

0, 𝑛 > 0𝑀 − 12

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𝑤! =1𝑀Π" =

1𝑀 , 𝑛 ≤ 0𝑀 − 1

2

0, 𝑛 > 0𝑀 − 12

Where 𝑤! = 𝑤#, 𝑤$, 𝑤%, ⋯ , 𝑤&'$ are the time-domain filter weights.

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𝑤! =1𝑀Π" =

1𝑀 , 𝑛 ≤ 0𝑀 − 1

2

0, 𝑛 > 0𝑀 − 12


Convolution of the data with the filter weights, 𝑤!, gives a weighted sum of the input within Π& or in this case, an average, since all the weights are equal.

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𝑤! =1𝑀Π" =

1𝑀 , 𝑛 ≤ 0𝑀 − 1

2

0, 𝑛 > 0𝑀 − 12



The output is then, 𝑧! = 𝑦! ∗ 𝑤! = 9()'*

*

𝑦(𝑤!'(

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Consider the running mean filter, possible the most common FIR filter.

𝑤! =1𝑀Π" =

1𝑀 , 𝑛 ≤ 0𝑀 − 1

2

0, 𝑛 > 0𝑀 − 12



The output is then, 𝑧! = 𝑦! ∗ 𝑤! = 9()'*

*

𝑦(𝑤!'( =1𝑀 9

() +' &'$%

+&'$%

𝑦(

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For example, let 𝑀 = 5

𝑧 10 =𝑦 8 + 𝑦 9 + 𝑦 10 + 𝑦 11 + 𝑦 12

5

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𝑧 10 =𝑦 8 + 𝑦 9 + 𝑦 10 + 𝑦 11 + 𝑦 12

5

𝑧 11 =𝑦 9 + 𝑦 10 + 𝑦 11 + 𝑦 12 + 𝑦 13

5

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𝑧 10 =𝑦 8 + 𝑦 9 + 𝑦 10 + 𝑦 11 + 𝑦 12

5

𝑧 11 =𝑦 9 + 𝑦 10 + 𝑦 11 + 𝑦 12 + 𝑦 13

5

= 𝑦 10 +𝑦 13 − 𝑦 8

5

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𝑧 10 =𝑦 8 + 𝑦 9 + 𝑦 10 + 𝑦 11 + 𝑦 12

5

𝑧 11 =𝑦 9 + 𝑦 10 + 𝑦 11 + 𝑦 12 + 𝑦 13

5

= 𝑧 10 +𝑦 13 − 𝑦 8

5Very fast to implement in code.

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𝑧 10 =𝑦 8 + 𝑦 9 + 𝑦 10 + 𝑦 11 + 𝑦 12

5

𝑧 11 =𝑦 9 + 𝑦 10 + 𝑦 11 + 𝑦 12 + 𝑦 13

5

= 𝑧 10 +𝑦 13 − 𝑦 8

5Very fast to implement in code.

We use the DFT to see the frequency domain response of this filter.

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𝑍, = 𝑌, A 𝐷𝐹𝑇 𝑤!

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𝑍, = 𝑌, A 𝐷𝐹𝑇 𝑤! = 𝑌,1𝑀

9!) +' &'$

%

+&'$%

𝑒'-%.,!/0

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𝑍, = 𝑌, A 𝐷𝐹𝑇 𝑤! = 𝑌,1𝑀

9!) +' &'$

%

+&'$%

𝑒'-%.,!/0

The DFT of a boxcar

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𝑍, = 𝑌, A 𝐷𝐹𝑇 𝑤! = 𝑌,1𝑀

9!) +' &'$

%

+&'$%

𝑒'-%.,!/0

= 𝑌,1𝑀sin 0𝑚𝜋𝑘

𝑁sin 0𝜋𝑘

𝑁

The DFT of a boxcar

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𝑍, = 𝑌, A 𝐷𝐹𝑇 𝑤! = 𝑌,1𝑀

9!) +' &'$

%

+&'$%

𝑒'-%.,!/0


𝑁sin 0𝜋𝑘

𝑁

The DFT of a boxcar

The discrete equivalent of a sync function.

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𝑍, = 𝑌, A 𝐷𝐹𝑇 𝑤! = 𝑌,1𝑀

9!) +' &'$

%

+&'$%

𝑒'-%.,!/0


𝑁sin 0𝜋𝑘

𝑁

The DFT of a boxcar

The discrete equivalent of a sync function.

A low pass filter with a lot of ripple in the stop band.

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In general, an M-point FIR filter an be written in the frequency domain as,

𝑊, = 9!)#

&'$

𝑤!𝑒'-%.,!/0

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𝑊, = 9!)#

&'$

𝑤!𝑒'-%.,!/0

If 𝑤! is symmetric and real, it has linear phase response.

𝑊, = 𝑒'-%.,(&'$)/0 9!)'(&'$)/%

(&'$)/%

𝑤!𝑒'-%.,!/0

Phase shift Time shift

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𝑊, = 9!)#

&'$

𝑤!𝑒'-%.,!/0


𝑊, = 𝑒'-%.,(&'$)/0 9!)'(&'$)/%

(&'$)/%

𝑤!𝑒'-%.,!/0


= 𝑒'-%.,(&'$)/0 2 9!)$

(&'$)/%

𝑤! cos 2𝜋𝑘𝑛/𝑁 + 𝑤#

𝑤! is symmetric (even function)

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𝑊, = 9!)#

&'$

𝑤!𝑒'-%.,!/0


𝑊, = 𝑒'-%.,(&'$)/0 9!)'(&'$)/%

(&'$)/%

𝑤!𝑒'-%.,!/0


= 𝑒'-%.,(&'$)/0 2 9!)$

(&'$)/%

𝑤! cos 2𝜋𝑘𝑛/𝑁 + 𝑤#

𝑤! is symmetric (even function)

= 𝑃(𝑘) A 𝐴(𝑘)

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𝑊, = 𝑃, A 𝐴,

𝑃, is a linear phase shift just like the 15f phase shift from before

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𝑊, = 𝑃, A 𝐴,


Time shifts of each frequency component are all the same relative to each other.

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𝑊, = 𝑃, A 𝐴,



No distortion.

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𝑊, = 𝑃, A 𝐴,



No distortion.

𝐴, is the real valued frequency domain amplitude response of the filter.

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𝑊, = 𝑃, A 𝐴,



No distortion.


The matlab conv(𝒙,𝒘) command is useful to convolve the filter sequence (or weights), 𝑤 with the input sequence (or data), 𝑥.

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𝑊, = 𝑃, A 𝐴,



No distortion.



This will produce an extra 𝑀 − 1 points on the output which is the time it takes the filter response to decay after the last data point.

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𝑊, = 𝑃, A 𝐴,



No distortion.



This will produce an extra 𝑀 − 1 points on the output which is the time it takes the filter response to decay after the last data point.

Just cut the extra points if not needed, y=y(1:N);

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The matlab filter command may also be used to implement your FIR filter. It’s designed for IIR filters (more later) which uses two vectors but works just as well with 1 vector.

y=filter(w,[1],x);

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One method of FIR filter design is to begin with an ideal continuous frequency domain response, Ω 𝑓 , and inverse FT on the Nyquist interval to get the time domain FIR weights.

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𝑤! = V'3!/%

3!/%Ω(𝑓)𝑒-%.3!𝑑𝑓 where 𝑓4 is the sample rate.

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𝑤! = V'3!/%


The problem here is there may be an infinite number of non-zero weights, 𝑤!.

It is frequently more intuitive to design a filter in the frequency domain.

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Consider the ideal low pass filter, a boxcar.

Ω 𝑓 = Π𝛼𝑓2

𝑓 = 0

Ω(𝑓)

𝑓 = −1𝛼

1𝛼

1

0Transitions at 53

%= ± $

%

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𝑓 = 0

Ω(𝑓)

𝑓 = −1𝛼

1𝛼

1

0Transitions at 53

%= ± $

%

𝑤! = V'3!/%

3!/%Ω(𝑓)𝑒-%.3!𝑑𝑓

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𝑓 = 0

Ω(𝑓)

𝑓 = −1𝛼

1𝛼

1

0Transitions at 53

%= ± $

%

𝑤! = V'3!/%

3!/%Ω(𝑓)𝑒-%.3!𝑑𝑓

For simplicity, let 𝑓4 = 1= V

'3!/5

3!/5𝑒-%.3!𝑑𝑓

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𝑓 = 0

Ω(𝑓)

𝑓 = −1𝛼

1𝛼

1

0Transitions at 53

%= ± $

%

𝑤! = V'3!/%

3!/%Ω(𝑓)𝑒-%.3!𝑑𝑓


'3!/5

3!/5𝑒-%.3!𝑑𝑓

= V'$/5

$/5𝑒-%.3!𝑑𝑓

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𝑓 = 0

Ω(𝑓)

𝑓 = −1𝛼

1𝛼

1

0Transitions at 53

%= ± $

%

𝑤! = V'3!/%

3!/%Ω(𝑓)𝑒-%.3!𝑑𝑓


'3!/5

3!/5𝑒-%.3!𝑑𝑓

= V'$/5

$/5𝑒-%.3!𝑑𝑓= 2V

#

$/5cos 2𝜋𝑓𝑛 𝑑𝑓

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𝑓 = 0

Ω(𝑓)

𝑓 = −1𝛼

1𝛼

1

0Transitions at 53

%= ± $

%

𝑤! = V'3!/%

3!/%Ω(𝑓)𝑒-%.3!𝑑𝑓


'3!/5

3!/5𝑒-%.3!𝑑𝑓

= V'$/5

$/5𝑒-%.3!𝑑𝑓= 2V

#

$/5cos 2𝜋𝑓𝑛 𝑑𝑓=

2𝛼 𝑠𝑖𝑛𝑐

2𝑛𝛼

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𝑤! =%5𝑠𝑖𝑛𝑐 %!

5, are the time domain weights of our filter for the ideal low pass

response.

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𝑤! =%5𝑠𝑖𝑛𝑐 %!


response.

The weights decay as $!

and only goes to 0 in the limit as 𝑛 → ∞.

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𝑤! =%5𝑠𝑖𝑛𝑐 %!


response.



We can truncate the weights using a boxcar from ± 0(&'$)%, but as we

found with windowing in our discussion on power spectra, this creates spectral leakage in the frequency domain by convolving the ideal filter, Ω(𝑓) with a sinc function.

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𝑤! =%5𝑠𝑖𝑛𝑐 %!


response.



We can truncate the weights using a boxcar from ± 0(&'$)%, but as we

found with windowing in our discussion on power spectra, this creates spectral leakage in the frequency domain by convolving the ideal filter, Ω(𝑓) with a sinc function.

This leads us right back to windowing.

Run matlab program aster_fig54ff.m

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We designed our FIR filter by first selecting the desired ideal frequency response, Ω(𝑓), then converting it to the discrete time domain with the integral transform,

𝑤! = V'3!/%


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𝑤! = V'3!/%


We then windowed 𝑤! to the desired number of weights (128 points in our example).

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𝑤! = V'3!/%


We then windowed 𝑤! to the desired number of weights (128 points in our example).

This is not the best way to design a FIR filter but it is the easiest.

Run matlab program firpmdemo.m

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Consider how successive superpositions of precisely aligned (think phase) cosines sum to create a discontinuity.

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Long periods are needed to make the flat parts and high frequencies are needed to construct the discontinuity.

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This leads us to a phenomena known as Gibbs Phenomena which is a result of constructing discontinuous signals with a finite number of discrete frequencies.


Long periods are needed to make the flat parts and high frequencies are needed to construct the discontinuity.

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On the one hand, we need the lower frequencies to construct the flat portion of the function, and high frequencies aligned to create the discontinuity.

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This creates overshoot at the discontinuity. We can make the overshoot more narrow by adding additional higher frequencies but the amplitude of the overshoot remains at 9%. This is Gibb’s phenomena.

Run gibbs.m

FIR Filters - University of MemphisFIR →Finite Impulse Response Finite duration time-domain impulse response, often ten to a few hundred points, applied with direct time-domain convolution.

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