1. CS 591 S1 – Computational Audio -- Spring, 2017
Wayne Snyder Computer Science Department Boston University
Lecture 7
Smoothing and Digital Filters
Subtractive synthesis
Formant synthesis of voice signals
Demo of synthesis platforms
Lecture 8 (Thursday)
Physical Modeling Synthesis
Karplus-Strong String Synthesis
HW 03 out tomorrow, due Sunday 2/26.
First Midterm is Thursday 3/3.
Book Report over Spring Break. 1

2. Smoothing Algorithms and Digital Filters
Smoothing refers to the processing of reducing small variations in a signal (or, more generally, a function) and emphasizing longer-term trends; in signal processing, this is usually cast more generally in terms of a filter, which supresses particular frequencies (e.g., higher frequencies) in a signal: 2

3. Smoothing Algorithms and Digital Filters
Smoothing Algorithms: We may think of smoothing as the process of transforming a signal using a weighted moving average of 2k+1 values around a sample for some k:
Original signal X:
…. X[i–k], … , X[i-1], X[i], X[i+1], … , X[i+k], ….
Ci-k * X[i–k] + … + Ci-1 * X[i-1] + Ci * X[i] + Ci+1 * X[i+1] + … + Ci+k * X[i+k]
= Y[i]
We “slide” this “smoothing window” through the original signal X (which has been padded with k 0’s at the beginning and end, to account for what to do with partial windows at the beginning and end) to produce a smoothed signal Y:
…. Y[i–k], … , Y[i-1], Y[i], Y[i+1], … , Y[i+k], ….
Note: To keep the amplitude of the smoothed signal in the same range, we assume that
Ci-k + … + Ci + …. + Ci+k = 1.0 3

4. Smoothing Algorithms and Digital Filters
Depending on how we set k (the width of the “smoothing window”) and the coefficients
Ci-k, …, Ci, …., Ci+k, we get different kinds of smoothing:
Rectangular (“Boxcar”) Smoothing: Each C = 1/(2k+1)
Triangular Smoothing: The C’s “swell” in a triangle, emphasizing the current sample and its near neighbors:
2k+1 = 7 4

5. Smoothing Algorithms and Digital Filters
This is often only done with “past” values of the current sample:
Exponential Smoothing uses an exponential (or geometric) decrease in the coefficients as you go farther back in the “past”:
α = 0.5 5

6. Smoothing Algorithms and Digital Filters
Examples:
Original Signal:
Rectangular Smoothing (k = 1):
Rectangular Smoothing (k = 2):
Larger windows mean more smoothing! 6

7. Smoothing Algorithms and Digital Filters
Examples:
Original Signal:
Rectangular Smoothing [0.2, 0.2, 0.2, 0.2, 0.2]:
Rectangular Smoothing [0.2, 0.2, 0.2, 0.0, 0.0]:
Exponential Smoothing [ , , 0.125, 0.25, 0.5, 0.0, 0.0, 0.0, 0.0 ]: 7

8. Smoothing Algorithms and Digital Filters
A similar technique which does not exactly fit this “weighted moving average” scheme is
median smoothing:
Y[i] = median( X[i-k], …., X[i], ….., X[i+k] )
This will ignore extreme outliers:
But otherwise acts similarly with rectangular smoothing, but with faster transitions to plateaus: 8

9. Smoothing Algorithms and Digital Filters
A useful variation of smoothing (call it “anti-smoothing” is to subtract the smoothed signal from the original; in the case of median smoothing, this will preserve the peaks and ignore the valleys! It can be useful in “peak picking” stages in audio programming: 9

10. Smoothing Algorithms and Digital Filters
In digital signal processing, the general process of modifying an input signal (e.g., by smoothing) is referred to as filtering:
Filters can be used to do a variety of tasks:
Amplitude modification: reduction, increase of amplitude
Frequency modification:
Low-pass filter (allow only frequencies below some threshold)
High-pass filter (allow only frequencies above some threshold)
Band-pass filter (allow only frequencies in some band)
Many miscellaneous tasks can be thought of as filtering:
Noise reduction
Tone controls
Reverberation
Vowel formation in vocal track 10

11. Digital Audio: Synthesis Concluded: Filters
We will consider only two classes of filters, those for reverb and for low- and band-pass filtering.
Here is the frequency response of an ideal low-pass filter, which filters out all frequencies above a certain cutoff fc: 11

12. Digital Audio: Synthesis Concluded: Filters
Realistically, however, no filter is ideal, and has a certain envelope for how it restricts frequencies; here is a realistic low-pass filter: 12

13. Digital Audio: Synthesis Concluded: Filters
Implementing Digital Filters
A Order-K Linear (Digital) Filter is a module that takes a sequence of samples x[0..N-1] and produces an output sequence y[0..N-1]:
The filter has a memory for K previous values for x[], and can do the following operations:
-- Add any two values
-- Multiply any value by a constant
Thus, the output is a linear function of the last K values of the input; a fifth-order filter would be defined by parameters A .. E:
y[i] = A*x[i] + B*x[i-1] + C*x[i-2] + D*x[i-3] + E*x[i-4]
y[i]
y[i]
Filter 13

14. Digital Audio: Synthesis Concluded: Filters
Let’s investigate the behavior of this for various values of the parameters; here is the baseline signal with equal values for all frequencies: y[i] = x[i] 14

15. Digital Audio: Synthesis Concluded: Filters
Let’s investigate the behavior of this for various values of the parameters; here we have created an amplitude reducer with even frequency response 15

16. Digital Audio: Synthesis Concluded: Filters
Let’s investigate the behavior of this for various values of the parameters; here we are summing the last two samples, which creates a typical low-pass filter: 16

17. Digital Audio: Synthesis Concluded: Filters
Let’s investigate the behavior of this for various values of the parameters; Here is a pass-band filter: 17

18. Digital Audio: Synthesis Concluded: Filters
Let’s investigate the behavior of this for various values of the parameters; another pass-band filter: 18

19. Digital Audio: Synthesis Concluded: Filters
Let’s investigate the behavior of this for various values of the parameters; Another pass-band filter: 19

20. Digital Audio: Synthesis Concluded: Filters
What is happening here??
Well, when we add samples at different points in the waveform, we may reinforce the wave form by constructive interference. Example: SR = 44100, we add samples at distance 100 (e.g., Y[1000] = 0.5*X[1000] + 0.5*X[900], Y[1001] = 0.5*X[1001] + 0.5*X[901], etc.) and a component sine wave is at frequency 44100/100 = 441 Hz, so that the delayed wave is perfectly in phase, and the component wave will be effectively the same:
X[900] X[1000] 20

21. Digital Audio: Synthesis Concluded: Filters
But when the component wave is exactly out of phase, we may remove the wave form by destructive interference. Example: SR = 44100, we add samples at distance 100 (e.g., Y[1000] = 0.5*X[1000] + 0.5*X[900], Y[1001] = 0.5*X[1001] + 0.5*X[901], etc.) and a component sine wave is at frequency 44100/200 = Hz, so that the delayed wave is out of phase by π, and the component wave will be removed, since the sum of each pair of points is 0.0.
X[900] X[1000] 21

22. Digital Audio: Synthesis Concluded: Filters
Reverb Filters
Using the same paradigm at a longer distance (e.g., 0.5 seconds) results in an echo or reverberation of the signal: 22