Intro to Sinusoids
• What is a sinusoid?
» Mathematical equation
x(t )  A cos( 2ft   )
t
x(t ) : Function of the time variable
A : Amplitude
f : Frequency (# cycles per sec, Hertz)

: Phase
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SINUSOIDAL SIGNAL
Acos( t  )
• FREQUENCY

– Radians/sec
– Hertz (cycles/sec)
 (2 ) f
• AMPLITUDE
– Magnitude
• PERIOD (in sec)
1 2
T 
f

• PHASE
Courtesy of James McClellan et al, Signal Processing First, 2003
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
A
Intro to Sinusoids
• What is a sinusoid?
» Plot
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Intro to Sinusoids
• Examples
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Intro to Sinusoids
• Examples
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Intro to Sinusoids
• Examples
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Intro to Sinusoids
• How to plot a sinusoid?
» Determine height
» Determine zero crossings
• Max height is +A
» peaks oscillate: +A and -A
» peaks occur when argument of cosine is a
multiple of  , i.e., 0,  ,2 ,3 ,...
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Intro to Sinusoids
• Zero crossings
» where plot crosses zero
» located at odd multiples of
 / 2,3 / 2,...
 / 2 , i.e.,
• Note: cycles repeat (periodic function)
» sufficient to determine peak and zero
crossings in one cycle and then replicate
cycle
» Period = length of cycle = T  1 / f
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Intro to Sinusoids
• We can also get equation from plot
» find the parameters A, f , and 
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Intro to Sinusoids
• We can also get equation from plot
» find the parameters A ,
f, and 
Answer:
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A  15, f  1000 Hz ,   0
Intro to Sinusoids
• Procedure to get equation from plot
• Find the amplitude A
• Find the period T by measuring the time
distance between positive peaks: f  1 / T
• Find the phase  by measuringt the “time
1
shift” (delay from 0) t1 of a peak. Note: t1
positive if peak moved to right, and
negative if peak moved to left.
• Then, compute   2ft1
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Intro to Sinusoids
• Sinusoids can be interpreted as a
“rotating phasor”
x(t )  cos( 2ft   )



•  (t ) rotating angle
 (t )
» f : speed of rotation (# cycles per
sec)
»  : initial angle start point
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Generating Signals using Sinusoids
• Main reason why sinusoids are
important: they are basic tools for
constructing other useful and more
complicated signals
• A new signal can be generated by
combining together different sinusoids
x(t )  A0  A1 cos( 2f1t  1 ) 
A2 cos( 2f 2 t   2 )  ...  AN cos( 2f N t   N )
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Generating Signals using Sinusoids
• Example: Beat Note Waveform
• Beat notes generated by adding two
sinusoids with nearby frequencies
• They can also be generated by playing two
neighboring piano keys
• Mathematically:
x(t )  A cos( 2f1t )  A cos( 2f 2 t )
where f1  f 0  d ; f 2  f 0  d ; d small.
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Generating Signals using Sinusoids
• Example: Beat Note Waveform
• Let
f o  200Hz ; d  20Hz  f1  220Hz ; f 2  180Hz
• Matlab Script File: beatnote.m
dur = 2.0;
fs = 8000;
f1 = 220; f2 = 180;
t = 0: (1/fs): dur;
x = cos(2*pi*f1*t)+cos(2*pi*f2*t);
plot(t,x); xlabel(‘Time,t’); ylabel(‘Beat Signal’);
sound(x,fs);
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Generating Signals using Sinusoids
• Example: Beat Note Waveform
• Let
f o  200Hz ; d  20Hz  f1  220Hz ; f 2  180Hz
• Matlab Function: beat.m
function [x,t] = beat(f0, d, dur);
fs = 8000;
f1 = f0+d; f2 = f0-d;
t = 0: (1/fs): dur;
x = cos(2*pi*f1*t)+cos(2*pi*f2*t);
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Matlab Functions
• Example: Beat Note Waveform
• Let
f o  200Hz ; d  20Hz  f1  220Hz ; f 2  180Hz
• Matlab Script File using Function beat.m
[x,t] = beat(200,20,1);
plot(t,x); xlabel(‘Time,t’); ylabel(‘Beat Signal’);
sound(x,fs);
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Generating Signals using Sinusoids
• A new signal can be generated by combining
together different sinusoids
x(t )  A0  A1 cos( 2f1t  1 ) 
A2 cos( 2f 2 t   2 )  ...  AN cos( 2f N t   N )
• Periodic signals obtained when frequencies
are integer multiples of f 0 :
f k  kf0
where f k is called harmonic frequency of f 0
and f 0 is called the fundamental
frequency.
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Generating Signals using Sinusoids
• A new periodic signal can be generated
by combining together sinusoids having
harmonically related frequencies
x(t )  A0  A1 cos( 2f 0 t  1 ) 
A2 cos( 2 2 f 0 t   2 )  ... 
AN cos( 2Nf 0 t   N )
• Period = length of one cycle = 1 / f 0
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PERIODIC SIGNALS
• Repeat every T secs
– Definition
– Example:
x (t )  x (t  T )
x(t )  cos (3t )
2
T ?
T
– Speech can be “quasi-periodic”
Courtesy of James McClellan et al, Signal Processing First, 2003
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2
3
T

3
Generating Signals using Sinusoids
• Example: Square Wave
N
8
x(t )  
cos( 2 (2m  1) f 0 t   / 2)
m  0 ( 2m  1)
• For N =1:
8
8
x(t )  cos( 2f o t   / 2)  cos( 2 3 f o t   / 2)

3
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Generating Signals using Sinusoids
Square Wave generates using only first 3 harmonic
Frequencies:
Courtesy of James McClellan et al, Signal Processing First, 2003
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T=0.1
Generating Signals using Sinusoids
• Example: Square Wave
• Matlab Function square.m
function [x,t] = square(f0, N, dur);
fs = 8000;
t = 0:1/fs:dur;
x = zeros(1,length(t));
for m = 0:N
x = x + (8/((2*m+1)*pi))*cos(2*pi*(2*m+1)*f0*t- pi/2);
end
• Generate and plot square waves with
f 0 = 25Hz, dur = 0.12 sec, and N=1,2,3,25
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Generating Signals using Sinusoids
• Example: Synthetic Vowel
f 0  100 Hz
A1  12,226, 1  1.508;
A4  29,416,  4  1.876; A5  48,836,  5  0.185;
A16  13,621, 16  1.449; A17  4,723, 17  0
• Generated signal approximates the
waveform produced by a man speaking
the sound “ah”.
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Vowel Waveform (sum of all 5 components)
Courtesy of James McClellan et al, Signal Processing First, 2003
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Generating Signals using Sinusoids
• Example: Synthetic Vowel
• Matlab Script File
dur = 1; f = 100;
fs = 8000;
t = 0:1/fs:dur;
x = 12226 * cos(2*pi*2*f*t + 1.508);
x = x + 29416 * cos(2*pi*4*f*t + 1.876);
x = x + 48836 * cos(2*pi*5*f*t - 0.185);
x = x + 13621 * cos(2*pi*16*f*t - 1.449);
x = x + 4723 * cos(2*pi*17*f*t);
plot(t,x); xlabel(‘Time, t’); ylabel(‘Vowel’)
sound(x,fs)
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TUNING FORK EXAMPLE
•
•
•
•
“A” is at 440 Hertz (Hz)
Waveform is a SINUSOIDAL SIGNAL
Computer plot looks like a sine wave
This should be the mathematical
formula:
A cos( 2 ( 440)t   )
Courtesy of James McClellan et al, Signal Processing First, 2003
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TUNING FORK A-440 Waveform
T  8.15  5.85
 2.3 ms
Time (sec)
f  1/ T
 1000 / 2.3
 435 Hz
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SPEECH EXAMPLE
• More complicated signal (BAT.WAV)
• Waveform x(t) is NOT a Sinusoid
• Theory will tell us
– x(t) is approximately a sum of sinusoids
– FOURIER ANALYSIS
• Break x(t) into its sinusoidal components
– Called the FREQUENCY SPECTRUM
Courtesy of James McClellan et al, Signal Processing First, 2003
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Speech Signal: BAT
• Nearly Periodic in Vowel Region
– Period is (Approximately) T = 0.0065 sec
Courtesy of James McClellan et al, Signal Processing First, 2003
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DIGITIZE the WAVEFORM
• x[n] is a SAMPLED SINUSOID
– A list of numbers stored in memory
• Sample at 11,025 samples per second
– Called the SAMPLING RATE of the A/D
– Time between samples is
• 1/11025 = 90.7 microsec
• Output via D/A hardware (at Fsamp)
Courtesy of James McClellan et al, Signal Processing First, 2003
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STORING DIGITAL SOUND
• x[n] is a SAMPLED SINUSOID
– A list of numbers stored in memory
•
•
•
•
CD rate is 44,100 samples per second
16-bit samples
Stereo uses 2 channels
Number of bytes for 1 minute is
– 2 X (16/8) X 60 X 44100 = 10.584 Mbytes
Courtesy of James McClellan et al, Signal Processing First, 2003
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SINES and COSINES
• Always use the COSINE FORM
A cos(t   )
• Sine is a special case:
sin(  t )  cos( t  2 )

Courtesy of James McClellan et al, Signal Processing First, 2003
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Sinusoidal Synthesis
• Sinusoids with DIFFERENT Frequencies
– SYNTHESIZE by Adding Sinusoids
N
x(t )   Ak cos( 2 f k t   k )
k 1
• SPECTRUM Representation
– Graphical Form shows DIFFERENT Frequencies
• SPECTROGRAM Tool
– Shows how frequency content varies in function of
time
– MATLAB function is specgram.m
Courtesy of James McClellan et al, Signal Processing First, 2003
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SPECTROGRAM EXAMPLE
• Two Constant Frequencies: Beats
cos( 2 (660)t ) sin( 2 (12)t )
Courtesy of James McClellan et al, Signal Processing First, 2003
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Frequency is the vertical axis
Time-Varying Frequency Diagram
A-440
Time is the horizontal axis
Courtesy of James McClellan et al, Signal Processing First, 2003
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Sinusoidal Synthesis: Motivation
• Synthesize Complicated Signals
– Musical Notes
• Piano uses 3 strings for many notes
• Chords: play several notes simultaneously
– Human Speech
• Vowels have dominant frequencies
• Application: computer generated speech
– Can all signals be generated this way?
• Sum of sinusoids?
Courtesy of James McClellan et al, Signal Processing First, 2003
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Fur Elise WAVEFORM
Beat
Notes
Courtesy of James McClellan et al, Signal Processing First, 2003
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SIMPLE TEST SIGNAL
• C-major SCALE: stepped frequencies
– Frequency is constant for each note
IDEAL
Courtesy of James McClellan et al, Signal Processing First, 2003
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SPECTROGRAM of C-Scale
Sinusoids ONLY
From SPECGRAM
ANALYSIS PROGRAM
ARTIFACTS at Transitions
Courtesy of James McClellan et al, Signal Processing First, 2003
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Spectrogram of LAB SONG
Sinusoids ONLY
Analysis Frame = 40ms
ARTIFACTS at Transitions
Courtesy of James McClellan et al, Signal Processing First, 2003
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Time-Varying Frequency
• Frequency can change vs. time
– Continuously, not stepped
• FREQUENCY MODULATION (FM)
x(t )  cos( 2 f c t  v(t ))
• CHIRP SIGNALS
– Linear Frequency Modulation (LFM)
Courtesy of James McClellan et al, Signal Processing First, 2003
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VOICE
New Signal: Linear FM
• Called Chirp Signals (LFM)
QUADRATIC
– Quadratic phase
x(t )  A cos( t  2 f 0 t   )
2
• Freq will change LINEARLY versus time
– Example of Frequency Modulation (FM)
– Define “instantaneous frequency”
Courtesy of James McClellan et al, Signal Processing First, 2003
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INSTANTANEOUS FREQ
• Definition
x (t )  A cos( (t ))
d  (t )
 i (t )  dt
Derivative
of the “Angle”
• For Sinusoid:
x (t )  A cos( 2 f 0t   )
 (t )  2 f 0t  
 i ( t ) 
d  (t )
dt
Courtesy of James McClellan et al, Signal Processing First, 2003
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 2 f 0
Makes sense
INSTANTANEOUS FREQ of the Chirp
• Chirp Signals have Quadratic phase
• Freq will change LINEARLY versus time
x (t )  A cos( t   t   )
2
  (t )   t   t  
2
 i ( t ) 
d  (t )
dt
Courtesy of James McClellan et al, Signal Processing First, 2003
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 2 t  
CHIRP SPECTROGRAM
Courtesy of James McClellan et al, Signal Processing First, 2003
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CHIRP WAVEFORM
Courtesy of James McClellan et al, Signal Processing First, 2003
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OTHER CHIRPS
 (t) can be anything:
x(t )  A cos( cos(  t )   )
 (t) could be speech or music:
– FM radio broadcast
 i ( t ) 
d  (t )
dt
Courtesy of James McClellan et al, Signal Processing First, 2003
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  sin(  t )
SINE-WAVE FREQUENCY MODULATION (FM)
Courtesy of James McClellan et al, Signal Processing First, 2003
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Music Synthesis
• Musical notes synthesized using a sinewave at
a given frequency.
• Musical scale is divided into 8 octaves; each
octave consists of 12 notes.
• Notes in each octave are related to notes in
previous and next octave:
– The frequency of a note is twice the frequency of
the corresponding note in the previous adjacent
(lower) octave.
• Each octave contains 12 notes (5 black keys
and 7 white) and the ratio between the
frequencies of the notes is constant between
successive notes: fnext note = 21/12 fprevious note
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Music - Octave
•D4 denotes D note in 4th octave
•A4 or A-440 note (tone at 440 Hz) is usually a reference note
called middle A.
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Music - Octave
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Music - Notation
4
• Musical notation shows which notes are to be played
and their relative timing
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Sinewave Synthesis
• Methods used
– Recursive
– Taylor’s Series
– Look-up Table
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Recursive Method
sin( A  B)  sin A cos B  cos A sin B
cos( A  B)  cos A cos B  sin A sin B
Let A =  and B =n where  is the angle and n the index
sin(   n )  sin(  [1  n])  sin  cos n  cos  sin n
cos(  n )  cos( [1  n])  cos  cos n  sin  sin n
For  = 1 and n= 1
sin( 1[1  1])  sin( 1) cos(1)  cos(1) sin( 1)
cos(1[1  1])  cos(1) cos(1)  sin( 1) sin( 1)
Sin(2) and cos(2) can be computed using only four multiplications and 2
addition/subtraction
Sin(1) and cos(1) must first be precomputed and stored in memory
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Recursive Method – Practical Example
• Generate a sine wave of frequency F = 100 Hz
having a sample frequency Fs equals to 8000
samples/s.
• Calculate the angle increment 
– Total number of samples/period = Fs/F = 80
– The angle increment  = 360/80 = 4.5
• Precompute sin(4.5) and cos(4.5)
• Apply the recursive method
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Taylor’s Series
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Look-up Table
• Calculate the values of L evenly spaced points on a
sinusoid and store them in memory
Lookup Values for L = 16
1
• Suppose L = 16 and Fs =
16000 samples/s
• The total number of cycles
obtained per second
is:16000/16 = 1000 Hz
0.8
0.6
Amplitude
0.4
0.2
0
-0.2
-0.4
-0.6
-0.8
-1
0
5
10
Sample
Base Address
0
Base Address + 1
sin(2/L)
Base Address + 2
sin(4/L)
.
.
.
Base Address + L – 1
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.
.
.
sin(2(L-1)/L)
15
– The frequency calculated is
called fundamental frequency
given by:
f fund 
1
1 F
x  S ( Hz )
L
L samples T
Look-up Table (Cont’d)
• If the samples are not read sequentially from
the look-up table (e.g., skipping one sample),
the frequency generated will be:
1
1 FS
f  '
x  ' ( Hz )
L samples T L
Example:
Suppose Fs = 16 Khz, L = 16, and  =
2:
f sin 

2 16000
FS 
 2000 Hz
L
16
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where:
L
L

'
and  is the
lookup table
increment index.
The maximum value 
can reach is bounded by
the Nyquist rate:
= L/2 => fsin= Fs/2;
Look-up Table (Cont’d)
• Frequency synthesized for different values of 
(L = 16)
 Frequency Generated (Hz)
What if  not integer?
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1
1000
2
2000
3
3000
4
4000
5
5000
6
6000
7
7000
8
8000
Maximum Frequency
Look-up Table – Non Integer 
• Two Methods
– Round – down
– Interpolation
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Look-up Table – Round Down Method
1.5
“+”: Look-up Table
“o”: Desired Output
1
L=8
0.5
 = 2.5
0
-0.5
-1
-1.5
0
1
2
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3
4
5
6
7
8
High Distortion!!
Look-up Table – Interpolation
Linear interpolation method used
y = mx + b
where:
m = line slope given by (Base_address + l +1) – (Base_address + l )
Base_address is the beginning address of the lookup table.
b = (Base_address + l ).
x = fractional part of the pointer with 0<x<1
1.5
y = linear approximated output sample.
(2,1)
Note that () denotes indirect addressing.
Interpolation
0.5.(0.7071-1)+1 =
0.8536
1
(3,0.7071)
0.5
0
-0.5
-1
-1.5
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0
1
2
3
4
5
6
7
8
Methods - Comparison
Look-up Table
Taylor’s Series
Recursive Method
Memory Space
Yes
No
No
Speed
Fast
Slow
Slow
Complexity
Low
High
Moderate
Accuracy
Moderate
High
High
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Amplitude Modulation of Tones - ADSR envelope
• Smoothing may be needed since the musical note will
not reach its full intensity instantaneously
• Attack – Decay – Sustain- Release (ADSR) envelope
- Attack time: time during which the musical note
reaches its peak
- Decay time: time required for the tone’s intensity to
partially die away
- Sustain time: time where the tone’s intensity remains
unchanged
- Release time: time for final attenuation
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ADSR - Examples
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ADSR Generation
- One segment at a time (A – D – S – R)
- Approximated as rising/decaying exponentials
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ADSR Generation – cont’d
- If Fs = 16000 samples/s and assuming a max
note duration of 1 second, we need to store 16K
in memory for the ADSR envelope values, which
consumes a lot of memory
- An alternative to the lookup ADSR is to try to
compute the values “on the fly” using the
equation:
y[n]  g.x(n)  (1^  g ). y[n  1]
where g is the rise/decay rate and x(n) is the desired
target value; initially x(-1)= 0.
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ADSR Generation – cont’d
- The equation presented previously is a firstorder IIR filter given by the following difference
equation:
y[n]  g.x[n]  (1  g ). y[n  1]
x[n]
g
y[n]

z-1
1-g
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ADSR Generation – cont’d
Why???
IIR Filter
Solve for the equation and show that the
output have an exponential shape
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