Transcript CS1371 Introduction to Computing for Engineers

CMPS1371
Introduction to Computing
for Engineers
PROCESSING SOUNDS
The Physics Of Sound
 Why do we hear what we hear?
 Sound is made when something vibrates.
 The vibration disturbs the air around it.
 This makes changes in air pressure.
 These changes in air pressure move through
the air as sound waves.
Sound Volume
 The louder a sound, the more energy it has.
This means loud sounds have a large
amplitude.
 Think about what an amplifier does: it makes
sounds louder. It is the amplitude that relates
to how loud sound is.
Sound Pitch
 All sound is made by things vibrating. The
faster things vibrate, the higher the pitch of the
sound produced.
 The vibrations being more frequent mean the
frequency of the wave increases.
Intensity Levels
Source
Decibels
Normal Breathing
Rustling Leaves
Soft Whisper
Mosquito
Quiet Office
Normal Conversation
Busy Street Traffic
Factory
Vacuum Cleaner
Train
Walkman at Maximum Level
Rock Concert
Machine Gun
Military Jet Takeoff
Rocket Engine
0
10
20
30
40
50
60
70
80
90
100
110
120
130
150
180
Description
Threshold of Hearing
Barely Audible
Quiet
Constant Exposure
Endangers Hearing
Threshold of Pain
Perforated Eardrum
Sound Recording and Playback
 Methods to store and reproduce sound is a
continual process for high quality



Phonograph
Magnetic tape
Digital recording
Record
 A sound will be collected as a vector
 The vector will provide signals over time to
represent the frequency (pitch) and amplitude
(intensity)
Sound Function
 SOUND function will play the vector as sound.
 sound(y,Fs) sends the signal in vector Y (with sample
frequency FS) out to the speaker on platforms that
support sound.

sound(y) plays the sound at the default sample rate
of 8192 Hz.

sound(y,Fs,bits) plays the sound using BITS
bits/sample if possible. Most platforms support
BITS=8 or 16.
Example:
load laughter
sound(y,Fs)
plot(y)
Read and Write Sound Files
 y = wavread(FILE)
 reads a wave file specified by the string FILE,
returning the sampled data in y
 wavwrite(y,Fs,NBITS,WAVEFILE)
 writes data Y to a Windows WAVE file
specified by the file name WAVEFILE, with a
sample rate of FS Hz and with NBITS number
of bits (default Fs = 8000 hz, NBITS = 16 bits)
 For audio files use:
 auread
 auwrite
Making Music with MATLAB
 Before we actually start making music, let's revise a few
AC waveform basics. Consider the sine wave shown in
the figure below:
 The sine wave shown here
can be described
mathematically as:
v = A sin(2π f t)
 where A is the Amplitude
(varying units), f is the
frequency (Hertz) and t is the
time (seconds).
 T is known as the time period
(seconds) and T=1/f
Music
 Sound waves are created when a waveform is
used to vibrate molecules in a material medium at
audio frequencies (300 Hz <= f <= 3 kHz).
 Example:

the MATLAB code to create a sine wave of
amplitude A = 1, at audio frequency of 466.16
Hz (corresponds to A#) would be:
>> v = sin(2*pi*466.16*[0:0.00125:1.0]);
Music
 Now, we can either plot this sine wave; or we
can hear it!!!
 To plot, simply type:
>> plot(v);
Music

To hear v, we need to convert the data to some
standard audio format

Matlab provides a function called wavwrite to
convert a vector into wav format and save it on
disk.
>> wavwrite(v, 'asharp.wav');
you can give any file name
Music
 Now, we can "play" this wav file called
asharp.wav using any multimedia player.

wavfunction returns 3 variables:



Vector signal
Sampling frequency
Number of bits
>> [y, Fs, bits] = wavread('asharp.wav');
>> sound(y, Fs)
Music
 Now that we can make a single note, we can
put notes together and make music!!!
 Let's look at the following piece of music:
AA
DD
EE
EE
C#C#
DD
F# F#
BB
C# C#
(repeat first two lines once)
EE
AA
B B (repeat once)
Music
 The American Standard Pitch for each of
these notes is:
A:
B:
C#:
D:
E:
F#:
440.00 Hz
493.88 Hz
554.37 Hz
587.33 Hz
659.26 Hz
739.99 Hz
Music
clear;
a=sin(2*pi*440*(0:0.000125:0.5));
b=sin(2*pi*493.88*(0:0.000125:0.5));
cs=sin(2*pi*554.37*(0:0.000125:0.5));
d=sin(2*pi*587.33*(0:0.000125:0.5));
e=sin(2*pi*659.26*(0:0.000125:0.5));
fs=sin(2*pi*739.99*(0:0.000125:0.5));
line1=[a,a,e,e,fs,fs,e,e,];
line2=[d,d,cs,cs,b,b,a,a,];
line3=[e,e,d,d,cs,cs,b,b];
song=[line1,line2,line3,line3,line1,line2];
wavwrite(song,'song.wav');
Sound
Pressure
Pressure
 SOUND:
 One dimensional function of changing airpressure in time
Time
t t
Time
Sound
 If the function is periodic, we perceive it as
sound with a certain frequency (else it’s noise).
Pressure
Pressure
 The frequency defines the pitch.
Time t
Time t
Sound
 The SHAPE of the curve defines the sound
character
String
String
Flute
Flute
Flute
Brass
Brass
Sound
 Listening to an orchestra, you can distinguish
between different instruments, although the
sound is a
Flute
Brass
String
SINGLE FUNCTION !
Sound
 If the sound produced by an
orchestra is the sum of different
instruments, could it be possible
that there are BASIC SOUNDS, that
can be combined to produce every
single sound ?
Sound
 The answer (Charles Fourier, 1822):
Any function that periodically repeats itself
can be expressed as the sum of sines/cosines
of different frequencies, each multiplied by
a different coefficient
Fourier
…A function…can be expressed as the sum
of sines/cosines…
What happens if we add sine and cosine ?
a * sin(ωt) + b * cos(ωt)
= A * sin(ωt + φ)
Adding sine and cosine of the same frequency yields just
another sine function with different phase and amplitude, but
same frequency.
Fourier
Any function that periodically repeats
itself…
 To change the shape of the function, we must
add sine-like functions with different
frequencies.
 As a formula:
f(x)= a0/2 + Σk=1..n akcos(kx) + bksin(kx)
Fourier Coefficients
Fourier
 The
set of ak, bk TOTALLY defines the
CURVE synthesized !
 We
can therefore describe the SHAPE of
the curve or the CHARACTER of the
sound by the (finite ?) set of FOURIER
COEFFICIENTS !
SAWTOOTH Function
Freq sin cos
f(x) = ½ - 1/π * Σn 1/n *sin (n*π*x)
1
1
0
2
1/2 0
3
1/3 0
4
1/4 0
The Problem
Given an arbitrary but periodically one
dimensional function (e.g. a sound), can
you tell the FOURIER COEFFICIENTS
to construct it ?
The answer (Charles Fourier):
Yes
Fast Fourier Transform
 MATLAB - function fft:
 Input: A vector, representing the discrete
function
 Output: The Fourier Coefficients as vector of
scaled imaginary numbers
 We can analyze the frequency content of
sound using the Fast Fourier Transform (fft)
Fast Fourier Transform
 "Fourier transform" goes from time domain to
the frequency domain
 Decompose a signal into it's sinusoids
Functionality of the fft
Examples