# Adding Sinusoids of the Same Frequency. Additive Synthesis. Spectrum. Music 270a: Modulation

Save this PDF as:

Size: px
Start display at page:

## Transcription

1 Adding Sinusoids of the Same Frequency Music 7a: Modulation Tamara Smyth, Department of Music, University of California, San Diego (UCSD) February 9, 5 Recall, that adding sinusoids of the same frequency but with possibly different amplitudes and phases, produces another sinusoid at that frequency: A n cos(ωt+φ n ) = n A n cosφ n cos(ωt) n = Bcos(ωt) Csin(ωt) = Acos(ωt+φ), where B = C = A n sinφ n sin(ωt) n A n cosφ n, n A n sinφ n, and where the amplitude and phase is given by A = ( ) C (B +C ), φ = tan. B n Music 7a: Modulation Spectrum Additive Synthesis When sinusoids of different frequencies are added together, the resulting signal is no longer sinusoidal. The spectrum of a signal is a graphical representation of the complex amplitudes, or phasors Ae jφ, of its frequency components (obtained using the DFT). The DFT of y( ) at frequency ω k is a measure of the the amplitude and phase of the complex DFT sinusoid e jω knt which is present in y( ) at frequency ω k : N Y(ω k ) = n= y(n)e jω knt Projecting a signal y( ) onto a DFT sinusoid provides a measure of how much that DFT sinusoid is in y( ). Discrete signals may be represented as the sum of sinusoids of arbitrary amplitudes, phases, and frequencies. Sounds may be synthesized by setting up a bank of oscillators, each set to the appropriate amplitude, phase and frequency: x(t) = A k cos(ω k t+φ k ) k= Since the output of each oscillator is added to produce the synthesized sound, the technique is called additive synthesis. Additive synthesis provides maximum flexibility in the types of sound that can be synthesized and can realize tones that are indistinguishable from the original. Signal analysis, which provides amplitude, phase and frequency functions for a signal, is often a prerequisite to additive synthesis, which is sometimes also called Fourier recomposition. Music 7a: Modulation 3 Music 7a: Modulation 4

2 Additive Synthesis Caveat Additive synthesis of standard periodic waveforms Drawback: it often requires many oscillators to produce good quality sounds, and can be very computationally demanding. Also, many functions are useful only for a limited range of pitch and loudness. For example, the timbre of a piano played at A4 is different from one played at A; the timbre of a trumpet played loudly is quite different from one played softly at the same pitch. It is possible however, to use some knowledge of acoustics to determine functions: e.g., in specifying amplitude envelopes for each oscillator, it is useful to know that in many acoustic instruments the higher harmonics attack last and decay first. Table : Other Simple Waveforms Synthesized by Adding Cosine Functions Type Harmonics Phase (cos) Phase (sin) square Odd, n = [,3,5,...,N] /n π/ triangle Odd, n = [,3,5,...,N] /n π/ sawtooth all, n = [,,3,...,N] /n π/ Figure : Summing sinusoids to produce other simple waveforms Music 7a: Modulation 5 Music 7a: Modulation 6 Harmonics and Pitch Beat Notes Notice that even though these new waveforms contain more than one frequency component, they are still periodic. Because each of these frequency components are integer multiples of some fundamental frequency f and producing a spectrum with evenly spaced frequency components, they are called harmonics. Signals with harmonic spectra have a periodic waveform where the period is the inverse of the fundamental. Pitch is our subjective response to a fundamental frequency within the audio range. The harmonics contribute to the timbre of a sound, but do not necessarily alter the pitch. What happens when two sinusoids that are not harmonically related are added? Magnitude Spectrum of Beat Note Frequency (Hz) Figure : Beat Note made by adding sinusoids at frequencies 8 Hz and Hz. The resulting waveform shows a periodic, low frequency amplitude envelope superimposed on a higher frequency sinusoid. Beat Note Waveform (f = Hz, f = Hz) The beat note comes about by adding two sinusoids that are very close in frequency. Music 7a: Modulation 7 Music 7a: Modulation 8

3 Multiplication of Sinusoids Modulation What happens when we multiply a low frequency sinusoids with a higher frequency sinusoid? sin(π()t) cos(π()t) ( e jπ()t e jπ()t )( e jπ()t +e jπ()t ) = j = ] [e jπ()t e jπ()t +e jπ(8)t e jπ(8)t 4j = [sin(π()t)+sin(π(8)t)] The multiplication of two sinusoids (as above) results in the sum of real sinusoids, and thus a spectrum having four frequency components (including the negative frequencies). Interestingly, none of the resulting spectral components are at the frequency of the multiplied sinusoids. Rather, they are at their sum and the difference. Sinusoidal multiplication can therefore be expressed as an addition (which makes sense because all signals can can be represented by the sum of sinusoids). Music 7a: Modulation 9 Modulation is the alteration of the amplitude, phase, or frequency of an oscillator in accordance with another signal. The oscillator being modulated is the carrier, and the altering signal is called the modulator. modulation, therefore, is the alteration of the amplitude of a carrier by a modulator. The spectral components generated by a modulated signal are called sidebands. There are three main techniques of amplitude modulation: Ring modulation Classical amplitude modulation Single-sideband modulation Music 7a: Modulation Ring Modulation Ring Modulation cont. Ring modulation (RM), introduced as the beat note waveform, occurs when modulation is applied directly to the amplitude input of the carrier modulator: x(t) = cos(πf t)cos(πf c t). Recall that this multiplication can also be expressed as the sum of sinusoids using the inverse of Euler s formula: x(t) = cos(πf t)+ cos(πf t), where f = f c f and f = f c +f. f f c f f f c f frequency Figure 3: Spectrum of ring modulation. Music 7a: Modulation f Notice again that neither the carrier frequency nor the modulation frequency are present in the spectrum. f f c f f f c f frequency Figure 4: Spectrum of ring modulation. Because of its spectrum, RM is also sometimes called double-sideband (DSB) modulation. Ring modulating can be realized without oscillators just by multiplying two signals together. The multiplication of two complex sounds produces a spectrum containing frequencies that are the sum and difference between each of the frequencies present in each of the sounds. The number of components in RM is two times the number of frequency components in one signal multiplied by the number of frequency components in the other. Music 7a: Modulation f

4 Classic Modulation RM and AM Spectra Classic amplitude modulation (AM) is the more general of the two techniques. In AM, the modulating signal includes a constant, a DC component, in the modulating term, x(t) = [A +cos(πf t)]cos(πf c t). Multiplying out the above equation yields x(t) = A cos(πf c t)+cos(πf t)cos(πf c t). The first term in the result above shows that the carrier frequency is actually present in the resulting spectrum. The second term can be expanded in the same way as was done for ring modulation, using the inverse Euler formula (left as an exercise). Where the centre frequency f c was absent in RM, it is present in classic AM. The sidebands are identical. f f A A f c f f f c f frequency Figure 5: Spectrum of amplitude modulation. f c f f f c f frequency Figure 6: Spectrum of ring modulation. A DC offset A in the modulating term therefore has the effect of including the centre frequency f c at an amplitude equal to the offset. f f Music 7a: Modulation 3 Music 7a: Modulation 4 RM and AM waveforms Single-Sideband modulation Because of the DC component, the modulating signal is often unipolar the entire signal is above zero and the instantaneous amplitude is always positive. 3 Unipolar signal Figure 7: A unipolar signal. Single-sideband modulation changes the amplitudes of two carrier waves that have quadrature phase relationship. Single-sideband modulation is given by x ssb (t) = x(t) cos(πf c t) H{x(t)} sin(πf c t), where H is the Hilbert transform. The effect of a DC offset in the modulating term is seen in the difference between AM and RM signals. 3 Modulation Ring Modulation Figure 8: AM (top) and RM (bottom) waveforms. The signal without the DC offset that oscillates between the positive and negative is called bipolar Music 7a: Modulation 5 Music 7a: Modulation 6

5 Frequency Modulation (FM) Chirp Signal Where AM is the modulation of amplitude, frequency modulation (FM) is the modulation of the carrier frequency with a modulating signal. Untill now we ve seen signals that do not change in frequency over time how do we modify the signal to obtain a time-varying frequency? One approach might be to concatenate small sequences, with each having a given frequency for the duration of the unit..5.5 Concatenating Sinusoids of Different Frequency Figure 9: A signal made by concatenating sinusoids of different frequencies will result in discontinuities if care is not taken to match the initial phase. Care must be taken in not introducing discontinuities when lining up the phase of each segment. Music 7a: Modulation 7 A chirp signal is one that sweeps linearly from a low to a high frequency. To produce a chirping sinusoid, modifying its equation so the frequency is time-varying will likely produce better results than concatenating segments. Recall that the original equation for a sinusoid is given by x(t) = Acos(ω t+φ) where the instantaneous phase, given by (ω t+φ), changes linearly with time. Notice that the time derivative of the phase is the radian frequency of the sinusoid ω, which in this case is a constant. More generally, if x(t) = Acos(θ(t)), the instantaneous frequency is given by ω(t) = d dt θ(t). Music 7a: Modulation 8 Chirp Sinusoid Sweeping Frequency Now, let s make the phase quadratic, and thus non-linear with respect to time. θ(t) = πµt +πf t+φ. The instantaneous frequency (the derivative of the phase θ), becomes which in Hz becomes ω i (t) = d dt θ(t) = 4πµt+πf, f i (t) = µt+f. Notice the frequency is no longer constant but changing linearly in time. To create a sinusoid with frequency sweeping linearly from f to f, consider the equation for a line y = mx+b to obtain instantaneous frequency: f(t) = f f T t+f, where T is the duration of the sweep. Music 7a: Modulation 9 The time-varying expression for the instantaneous frequency can be used in the original equation for a sinusoid,.5.5 x(t) = Acos(πf(t)t+φ). Chirp signal swept from to 3 Hz Time(s) Phase summary: Figure : A chirp signal from to 3 Hz. If the instantaneous phase θ(t) is constant, the frequency is zero (DC). If θ(t) is linear, the frequency is fixed (constant). if θ(t) is quadratic, the frequency changes linearly with time. Music 7a: Modulation

6 Vibrato simulation FM Vibrato Vibrato is a term used to describe a wavering of pitch. Vibrato (in varying amounts) occurs very naturally in the singing voice and in many sustained instruments (where the musician has control after the note has been played), such as the violin, wind instruments, the theremin, etc.). In Vibrato, the frequency does not change linearly but rather sinusoidally, creating a sense of a wavering pitch. Since the instantaneous frequency of the sinusoid is the derivative of the instantaneous phase, and the derivative of a sinusoid is a sinusoid, a vibrato can be simulated by applying a sinusoid to the instantaneous phase of a carrier signal: x(t) = A c cos(πf c t+a m cos(πf m t+φ m )+φ c ), that is, by FM synthesis. FM synthesis can create a vibrato effect, where the instantaneous frequency of the carrier oscillator varies over time according to the parameters controlling the width of the vibrato (the deviation from the carrier frequency) the rate of the vibrato. The width of the vibrato is determined by the amplitude of the modulating signal, A m. The rate of vibrato is determined by the frequency of the modulating signal, f m. In order for the effect to be perceived as vibrato, the vibrato rate must be below the audible frequency range and the width made quite small. Music 7a: Modulation Music 7a: Modulation FM Synthesis of Musical Instruments Frequency Modulation When the vibrato rate is in the audio frequency range, and when the width is made larger, the technique can be used to create a broad range of distinctive timbres. Frequency modulation (FM) synthesis was invented by John Chowning at Stanford University s Center for Computer Research in Music and Acoustics (CCRMA). FM synthesis uses fewer oscillators than either additive or AM synthesis to introduce more frequency components in the spectrum. Where AM synthesis uses a signal to modulate the amplitude of a carrier oscillator, FM synthesis uses a signal to modulate the frequency of a carrier oscillator. The general equation for an FM sound synthesizer is given by x(t) = A(t)[cos(πf c t+i(t)cos(πf m t+φ m )+φ c ], where A(t) the time varying amplitude f c the carrier frequency I(t) the modulation index f m the modulating frequency φ m,φ c arbitrary phase constants. Music 7a: Modulation 3 Music 7a: Modulation 4

7 Modulation Index Modulation Index cont. The function I(t), called the modulation index envelope, determines significantly the harmonic content of the sound. Given the general FM equation x(t) = A(t)[cos(πf c t+i(t)cos(πf m t+φ m )+φ c ], the instantaneous frequency f i (t) (in Hz) is given by f i (t) = d πdt θ(t) = d πdt [πf ct+i(t)cos(πf m t+φ m )+φ c ] = π [πf c I(t)sin(πf m t+φ m )πf m + d dt I(t)cos(πf mt+φ m )] = f c I(t)f m sin(πf m t+φ m )+ d πdt I(t)cos(πf mt+φ m ). Music 7a: Modulation 5 Without going any further in solving this equation, it is possible to get a sense of the effect of the modulation index I(t) from f i (t) = f c I(t)f m sin(πf m t+φ m )+ di(t) cos(πf m t+φ m )/π. dt We may see that if I(t) is a constant (and it s derivative is zero), the third term goes away and the instantaneous frequency becomes f i (t) = f c I(t)f m sin(πf m t+φ m ). Notice now that in the second term, the quantity I(t)f m multiplies a sinusoidal variation of frequency f m, indicating that I(t) determines the maximum amount by which the instantaneous frequency deviates from the carrier frequency f c. Since the modulating frequency f m is at audio rates, this translates to addition of harmonic content. Since I(t) is a function of time, the harmonic content, and thus the timbre, of the synthesized sound may vary with time. Music 7a: Modulation 6 FM Sidebands Bessel Functions of the First Kind The upper and lower sidebands produced by FM are given by f c ±kf m. The amplitude of the k th sideband is given by J k, a Bessel function of the first kind, of order k. magnitude J.5.5 J fc 4fm fc 3fm fc fm fc fm fc fc +fm fc +fm fc +3fm fc +4fm frequency J J Figure : Sidebands produced by FM synthesis. The modulation index may be see as I = d f m, where d is the amount of frequency deviation from f c. When d =, the index I is also zero, and no modulation occurs. Increasing d causes the sidebands to acquire more power at the expense of the power in the carrier frequency. Music 7a: Modulation 7 J J Figure : Bessel functions of the first kind, plotted for orders through 5. Notice that higher order Bessel functions, and thus higher order sidebands, do not have significant amplitude when I is small. Bessel functions are solutions to Bessel s differential equation. Music 7a: Modulation 8

8 Higher values of I therefore produce higher order sidebands. In general, the highest-ordered sideband that has significant amplitude is given by the approximate expression k = I +. Odd-Numbered Lower Sidebands The amplitude of the odd-numbered lower sidebands is the appropriate Bessel function multipled by -, since odd-ordered Bessel functions are odd functions. That is J k = J k. J.5.5 J J J J J Figure 3: Bessel functions of the first kind, plotted for odd orders. Music 7a: Modulation 9 Music 7a: Modulation 3 Effect of Phase in FM FM Spectrum The phase of a spectral component does not have an audible effect unless other spectral components of the same frequency are present, i.e., there is interference. In the case of frequency overlap, the amplitudes will either add or subtract, resulting in a perceived change in the sound. If the FM spectrum contains frequency components below Hz, they are folded over the Hz axis to their corresponding positive frequencies (akin to aliasing or folding over the Nyquist limit). In the case that the carrier is an odd function (a sine wave), the act of folding reverses the phase. A sideband with a negative frequency is equivalent to a component with the corresponding positive frequency but with the opposite phase. The frequencies present in an FM spectrum are f c ±kf m, where k is an integer greater than zero (the carrier frequency component is at k = ) Spectrum of a simple FM instrument: fc =, fm =, I = Frequency (Hz) Figure 4: Spectrum of a simple FM instrument, where fc =, fm =, and I = Spectrum of a simple FM instrument: fc = 9, fm = 6, I = Frequency (Hz) Figure 5: Spectrum of a simple FM instrument, where fc = 9, fm = 6, and I =. Music 7a: Modulation 3 Music 7a: Modulation 3

9 Fundamental Frequency in FM Missing Harmonics in FM Spectrum In determining the fundamental frequency of an FM sound, first represent the ratio of the carrier and modulator frequencies as a reduced fraction, f c = N, f m N where N and N are integers with no common factors. The fundamental frequency is then given by f = f c = f m. N N Example: a carrier frequency f c = and modulator frequency f m = yields the ratio of f c = f m = = N. N and a fundamental frequency of f = = =. Likewise the ratio of f c = 9 to f m = 6 is 3: and the fundamental frequency is given by f = 9 3 = 6 = 3. Music 7a: Modulation 33 For N >, every N th harmonic of f is missing in the spectrum. This can be seen in the plot below where the ratio of the carrier to the modulator is 4:3 and N = Spectrum of a simple FM instrument: fc = 4, fm = 3, I = Frequency (Hz) Figure 6: Spectrum of a simple FM instrument, where fc = 4, fm = 3, and I =. Notice the fundamental frequency f is and every third multiple of f is missing from the spectrum. Music 7a: Modulation 34 Some FM instrument examples When implementing simple FM instruments, we have several basic parameters that will effect the overall sound:. The duration,. The carrier and modulating frequencies 3. The maximum (and in some cases minimum) modulating index scalar 4. The envelopes that define how the amplitude and modulating index evolve over time. Using the information taken from John Chowning s article on FM (details of which appear in the text Computer Music (pp. 5-7)), we may develop envelopes for the following simple FM instruments: bell-like tones, wood-drum brass-like tones clarinet-like tones Bell Woodwind Brass Wood drum Env Mod. Index Figure 7: Envelopes for FM bell-like tones, wood-drum tones, brass-like tones and clarinet tones. Music 7a: Modulation 35 Music 7a: Modulation 36

10 Formants Formants with Two Carrier Oscillators Another characteristic of sound, in addition to its spectrum, is the presence of formants, or peaks in the spectral envelope. The formants describe certain regions in the spectrum where there are strong resonances (where the amplitude of the spectral components is considerably higher). As an example, pronounce aloud the vowels a, e, i, o, u while keeping the same pitch for each. Since the pitch is the same, we know the integer relationship of the spectral components is the same. The formants are what allows us to hear a difference between the vowel sounds. In FM synthesis, the peaks in the spectral envelop can be controlled using an additional carrier oscillator. In the case of a single oscillator, the spectrum is centered around a carrier frequency. With an additional oscillator, an additional spectrum may be generated that is centered around a formant frequency. When the two signals are added, their spectra are combined. If the same oscillator is used to modulate both carriers (though likely using seperate modulation indeces), and the formant frequency is an integer multiple of the fundamental, the spectra of both carriers will combine in such a way the the components will overlap, and a peak will be created at the formant frequency. Music 7a: Modulation 37 Music 7a: Modulation 38 Towards a Chowning FM Trumpet FM spectrum with first carrier: fc=4, fm=4, and f= FM spectrum with second carrier: fc=, fm=4, and f= FM spectrum using first and second carriers Frequency (Hz) Figure 8: The spectrum of individiual and combined FM signals. In Figure 8, two carriers are modulated by the same oscillator with a frequency f m. The index of modulation for the first and second carrier is given by I and I /I respectively. The value I is usually less than I, so that the ratio I /I is small and the spectrum does not spread too far beyond the region of the formant. The frequency of the second carrier f c is chosen to be a harmonic of the fundamental frequency f, so that it is close to the desired formant frequency f f, f c = nf = int(f f /f +.5)f. This ensures that the second carrier frequency remains harmonically related to f. If f changes, the scond carrier frequency will remain as close as possible to the desired formant frequency f f while remaining an integer multiple of the fundamental frequency f. Music 7a: Modulation 39 Music 7a: Modulation 4

11 Two Modulating Oscillators Just as the number of carriers can be increased, so can the number of modulating oscillators. To create even more spectral variety, the modulating waveform may consist of the sum of several sinusoids. If the carrier frequency is f c and the modulating frequencies are f m and f m, then the resulting spectrum will contain components at the frequency given by f c ±if m ±kf m, where i and k are integers greater than or equal to. For example, when f c = Hz, f m = Hz, and f m = 3 Hz, the spectral component present in the sound at 4 Hz is the combination of sidebands given by the pairs: i = 3,k = ;i =,k = ;i = 3,k = ; and so on (see Figure 9) FM spectrum with first modulator: fc= and fm= FM spectrum with second modulator: fc= and fm= FM spectrum with two modulators: fc=, fm=, and fm= Figure 9: The FM spectrum produced by a modultor with two frequency components. Music 7a: Modulation 4 Music 7a: Modulation 4 Two Modulators cont. Modulation indeces are defined for each component: I is the index that charaterizes the spectrum produced by the first modulating oscillator, and I is that of the second. The amplitude of the i th, k th sideband (A i,k ) is given by the product of the Bessel functions A i,k = J i (I )J k (I ). Like in the previous case of a single modulator, when i, k is odd, the Bessel functions assume the opposite sign. For example, if i = and k = 3 (where the negative superscript means that k is subtracted), the amplitude is A,3 = J (I )J 3 (I ). In a harmonic spectrum, the net amplitude of a component at any frequency is the combination of many sidebands, where negative frequencies foldover the Hz bin (Computer Music). Music 7a: Modulation 43

### Synthesis Techniques. Juan P Bello

Synthesis Techniques Juan P Bello Synthesis It implies the artificial construction of a complex body by combining its elements. Complex body: acoustic signal (sound) Elements: parameters and/or basic signals

### Voltage. Oscillator. Voltage. Oscillator

fpa 147 Week 6 Synthesis Basics In the early 1960s, inventors & entrepreneurs (Robert Moog, Don Buchla, Harold Bode, etc.) began assembling various modules into a single chassis, coupled with a user interface

### Lecture 4: Analog Synthesizers

ELEN E4896 MUSIC SIGNAL PROCESSING Lecture 4: Analog Synthesizers 1. The Problem Of Electronic Synthesis 2. Oscillators 3. Envelopes 4. Filters Dan Ellis Dept. Electrical Engineering, Columbia University

### SIGNAL PROCESSING & SIMULATION NEWSLETTER

1 of 10 1/25/2008 3:38 AM SIGNAL PROCESSING & SIMULATION NEWSLETTER Note: This is not a particularly interesting topic for anyone other than those who ar e involved in simulation. So if you have difficulty

### RightMark Audio Analyzer

RightMark Audio Analyzer Version 2.5 2001 http://audio.rightmark.org Tests description 1 Contents FREQUENCY RESPONSE TEST... 2 NOISE LEVEL TEST... 3 DYNAMIC RANGE TEST... 5 TOTAL HARMONIC DISTORTION TEST...

### Laboratory Assignment 4. Fourier Sound Synthesis

Laboratory Assignment 4 Fourier Sound Synthesis PURPOSE This lab investigates how to use a computer to evaluate the Fourier series for periodic signals and to synthesize audio signals from Fourier series

### Fourier Analysis. Nikki Truss. Abstract:

Fourier Analysis Nikki Truss 09369481 Abstract: The aim of this experiment was to investigate the Fourier transforms of periodic waveforms, and using harmonic analysis of Fourier transforms to gain information

### Chapter 8. Introduction to Alternating Current and Voltage. Objectives

Chapter 8 Introduction to Alternating Current and Voltage Objectives Identify a sinusoidal waveform and measure its characteristics Describe how sine waves are generated Determine the various voltage and

### Laboratory Manual and Supplementary Notes. CoE 494: Communication Laboratory. Version 1.2

Laboratory Manual and Supplementary Notes CoE 494: Communication Laboratory Version 1.2 Dr. Joseph Frank Dr. Sol Rosenstark Department of Electrical and Computer Engineering New Jersey Institute of Technology

### Using GNU Radio Companion: Tutorial 4. Using Complex Signals and Receiving SSB

Using GNU Radio Companion: Tutorial 4. Using Complex Signals and Receiving SSB This tutorial is a guide to receiving SSB signals. It will also illustrate some of the properties of complex (analytic) signals

### Introduction of Fourier Analysis and Time-frequency Analysis

Introduction of Fourier Analysis and Time-frequency Analysis March 1, 2016 Fourier Series Fourier transform Fourier analysis Mathematics compares the most diverse phenomena and discovers the secret analogies

### A Tutorial on Fourier Analysis

A Tutorial on Fourier Analysis Douglas Eck University of Montreal NYU March 26 1.5 A fundamental and three odd harmonics (3,5,7) fund (freq 1) 3rd harm 5th harm 7th harmm.5 1 2 4 6 8 1 12 14 16 18 2 1.5

### 7. Beats. sin( + λ) + sin( λ) = 2 cos(λ) sin( )

34 7. Beats 7.1. What beats are. Musicians tune their instruments using beats. Beats occur when two very nearby pitches are sounded simultaneously. We ll make a mathematical study of this effect, using

### Fast Fourier Transforms and Power Spectra in LabVIEW

Application Note 4 Introduction Fast Fourier Transforms and Power Spectra in LabVIEW K. Fahy, E. Pérez Ph.D. The Fourier transform is one of the most powerful signal analysis tools, applicable to a wide

### 10: FOURIER ANALYSIS OF COMPLEX SOUNDS

10: FOURIER ANALYSIS OF COMPLEX SOUNDS Amplitude, loudness, and decibels Several weeks ago we found that we could synthesize complex sounds with a particular frequency, f, by adding together sine waves

### L6: Short-time Fourier analysis and synthesis

L6: Short-time Fourier analysis and synthesis Overview Analysis: Fourier-transform view Analysis: filtering view Synthesis: filter bank summation (FBS) method Synthesis: overlap-add (OLA) method STFT magnitude

### Graham s Guide to Synthesizers (part 1) Analogue Synthesis

Graham s Guide to Synthesizers (part ) Analogue Synthesis Synthesizers were originally developed to imitate or synthesise the sounds of acoustic instruments electronically. Early synthesizers used analogue

### Lecture 8 ELE 301: Signals and Systems

Lecture 8 ELE 3: Signals and Systems Prof. Paul Cuff Princeton University Fall 2-2 Cuff (Lecture 7) ELE 3: Signals and Systems Fall 2-2 / 37 Properties of the Fourier Transform Properties of the Fourier

### A Sound Analysis and Synthesis System for Generating an Instrumental Piri Song

, pp.347-354 http://dx.doi.org/10.14257/ijmue.2014.9.8.32 A Sound Analysis and Synthesis System for Generating an Instrumental Piri Song Myeongsu Kang and Jong-Myon Kim School of Electrical Engineering,

### Angle Modulation, II. Lecture topics FM bandwidth and Carson s rule. Spectral analysis of FM. Narrowband FM Modulation. Wideband FM Modulation

Angle Modulation, II EE 179, Lecture 12, Handout #19 Lecture topics FM bandwidth and Carson s rule Spectral analysis of FM Narrowband FM Modulation Wideband FM Modulation EE 179, April 25, 2014 Lecture

### VCO Phase noise. Characterizing Phase Noise

VCO Phase noise Characterizing Phase Noise The term phase noise is widely used for describing short term random frequency fluctuations of a signal. Frequency stability is a measure of the degree to which

### where T o defines the period of the signal. The period is related to the signal frequency according to

SIGNAL SPECTRA AND EMC One of the critical aspects of sound EMC analysis and design is a basic understanding of signal spectra. A knowledge of the approximate spectral content of common signal types provides

### Frequency Domain Characterization of Signals. Yao Wang Polytechnic University, Brooklyn, NY11201 http: //eeweb.poly.edu/~yao

Frequency Domain Characterization of Signals Yao Wang Polytechnic University, Brooklyn, NY1121 http: //eeweb.poly.edu/~yao Signal Representation What is a signal Time-domain description Waveform representation

### 0. Basic Audio Terminology. Wave Concepts. Oscillator, Waveform and Timbre. Wave Graphs (time-amplitude domain) Harmonic Series (Harmonic Contents)

0. Basic Audio Terminology Wave Concepts The basic set of waveforms covered by this document are simple mathematical shapes which are found on analogue synthesizers. The first waveform to look at is the

### Analysis/resynthesis with the short time Fourier transform

Analysis/resynthesis with the short time Fourier transform summer 2006 lecture on analysis, modeling and transformation of audio signals Axel Röbel Institute of communication science TU-Berlin IRCAM Analysis/Synthesis

### Continuous Time Signals (Part - I) Fourier series

Continuous Time Signals (Part - I) Fourier series (a) Basics 1. Which of the following signals is/are periodic? (a) s(t) = cos t + cos 3t + cos 5t (b) s(t) = exp(j8 πt) (c) s(t) = exp( 7t) sin 1πt (d)

### Lecture 1-4: Harmonic Analysis/Synthesis

Lecture 1-4: Harmonic Analysis/Synthesis Overview 1. Classification of waveform shapes: we can broadly divide waveforms into the classes periodic (repeat in time) and aperiodic (don't repeat in time).

### FIR Filter Design. FIR Filters and the z-domain. The z-domain model of a general FIR filter is shown in Figure 1. Figure 1

FIR Filters and the -Domain FIR Filter Design The -domain model of a general FIR filter is shown in Figure. Figure Each - box indicates a further delay of one sampling period. For example, the input to

### Chapter 6. Single Sideband Transmitter Tests and Measurements

Chapter 6 Single Sideband Transmitter Tests and Measurements This chapter deals with the basic tests and measurements that can be used to evaluate the performance of single sideband suppressed carrier

### Curve Fitting. Next: Numerical Differentiation and Integration Up: Numerical Analysis for Chemical Previous: Optimization.

Next: Numerical Differentiation and Integration Up: Numerical Analysis for Chemical Previous: Optimization Subsections Least-Squares Regression Linear Regression General Linear Least-Squares Nonlinear

### 10: FOURIER ANALYSIS OF COMPLEX SOUNDS

10: FOURIER ANALYSIS OF COMPLEX SOUNDS Introduction Several weeks ago we found that we could synthesize complex sounds with a particular frequency, f, by adding together sine waves from the harmonic series

### Korg Volca FM Programming Guide

Korg Volca FM Programming Guide By Andrew Shakinovsky Introduction and Basics This guide goes over the details of programming the Korg Volca FM. It does not cover the basics of performing with the volca

### Computer Networks and Internets, 5e Chapter 6 Information Sources and Signals. Introduction

Computer Networks and Internets, 5e Chapter 6 Information Sources and Signals Modified from the lecture slides of Lami Kaya (LKaya@ieee.org) for use CECS 474, Fall 2008. 2009 Pearson Education Inc., Upper

### chapter Introduction to Digital Signal Processing and Digital Filtering 1.1 Introduction 1.2 Historical Perspective

Introduction to Digital Signal Processing and Digital Filtering chapter 1 Introduction to Digital Signal Processing and Digital Filtering 1.1 Introduction Digital signal processing (DSP) refers to anything

### Little LFO. Little LFO. User Manual. by Little IO Co.

1 Little LFO User Manual Little LFO by Little IO Co. 2 Contents Overview Oscillator Status Switch Status Light Oscillator Label Volume and Envelope Volume Envelope Attack (ATT) Decay (DEC) Sustain (SUS)

### TCOM 370 NOTES 99-2 FOURIER SERIES, BANDWIDTH, AND SIGNALING RATES ON DATA TRANSMISSION LINKS

TCOM 37 NOTES 99-2 FOURIER SERIES, BANDWIDTH, AND SIGNALING RATES ON DATA TRANSMISSION LINKS. PULSE TRANSMISSION We now consider some fundamental aspects of binary data transmission over an individual

### Fourier Analysis. A cosine wave is just a sine wave shifted in phase by 90 o (φ=90 o ). degrees

Fourier Analysis Fourier analysis follows from Fourier s theorem, which states that every function can be completely expressed as a sum of sines and cosines of various amplitudes and frequencies. This

### Trigonometric functions and sound

Trigonometric functions and sound The sounds we hear are caused by vibrations that send pressure waves through the air. Our ears respond to these pressure waves and signal the brain about their amplitude

### Mathematical Harmonies Mark Petersen

1 Mathematical Harmonies Mark Petersen What is music? When you hear a flutist, a signal is sent from her fingers to your ears. As the flute is played, it vibrates. The vibrations travel through the air

### Experiment 5 A Fourier series analyzer

Experiment 5 A Fourier series analyzer Achievements in this experiment Compose arbitrary periodic signals from a series of sine and cosine waves. Confirm the Fourier Series equation. Compute fourier coefficients

### Laboratory #2: AC Circuits, Impedance and Phasors Electrical and Computer Engineering EE University of Saskatchewan

Authors: Denard Lynch Date: Aug 30 - Sep 28, 2012 Sep 23, 2013: revisions-djl Description: This laboratory explores the behaviour of resistive, capacitive and inductive elements in alternating current

Measurement Lab Fourier Analysis Last Modified 9/5/06 Any time-varying signal can be constructed by adding together sine waves of appropriate frequency, amplitude, and phase. Fourier analysis is a technique

### 2 Background: Fourier Series Analysis and Synthesis

GEORGIA INSTITUTE OF TECHNOLOGY SCHOOL of ELECTRICAL and COMPUTER ENGINEERING ECE 2025 Spring 2001 Lab #11: Design with Fourier Series Date: 3 6 April 2001 This is the official Lab #11 description. The

### Frequency Domain Analysis

Exercise 4. Frequency Domain Analysis Required knowledge Fourier-series and Fourier-transform. Measurement and interpretation of transfer function of linear systems. Calculation of transfer function of

### Experiment 3: Double Sideband Modulation (DSB)

Experiment 3: Double Sideband Modulation (DSB) This experiment examines the characteristics of the double-sideband (DSB) linear modulation process. The demodulation is performed coherently and its strict

### Fourier Series and Integrals

Fourier Series and Integrals Fourier Series et f(x) be a piecewise linear function on [,] (This means that f(x) may possess a finite number of finite discontinuities on the interval). Then f(x) can be

### MTC191 / E.Walker Sound & Waveforms

MTC191 / E.Walker Sound & Waveforms Understanding how sound works will help us learn to manipulate it in the form of audio signals, such as an analog voltage or a digital signal that is stored and played

### The unique waveform control section starts with simple familiar waves and curves them into waves you haven't heard before.

The Q167 LFO++ is a Low Frequency Oscillator and universal modulator with a wide range of interesting features. This functionally-dense module creates complex modulation with a variety of common and unusual

### R f. V i. ET 438a Automatic Control Systems Technology Laboratory 4 Practical Differentiator Response

ET 438a Automatic Control Systems Technology Laboratory 4 Practical Differentiator Response Objective: Design a practical differentiator circuit using common OP AMP circuits. Test the frequency response

### TCOM 370 NOTES 99-4 BANDWIDTH, FREQUENCY RESPONSE, AND CAPACITY OF COMMUNICATION LINKS

TCOM 370 NOTES 99-4 BANDWIDTH, FREQUENCY RESPONSE, AND CAPACITY OF COMMUNICATION LINKS 1. Bandwidth: The bandwidth of a communication link, or in general any system, was loosely defined as the width of

### NOTES ON PHASORS. where the three constants have the following meanings:

Chapter 1 NOTES ON PHASORS 1.1 Time-Harmonic Physical Quantities Time-harmonic analysis of physical systems is one of the most important skills for the electrical engineer to develop. Whether the application

### Amplitude Modulation Fundamentals

3 chapter Amplitude Modulation Fundamentals In the modulation process, the baseband voice, video, or digital signal modifies another, higher-frequency signal called the carrier, which is usually a sine

### DDS. 16-bit Direct Digital Synthesizer / Periodic waveform generator Rev. 1.4. Key Design Features. Block Diagram. Generic Parameters.

Key Design Features Block Diagram Synthesizable, technology independent VHDL IP Core 16-bit signed output samples 32-bit phase accumulator (tuning word) 32-bit phase shift feature Phase resolution of 2π/2

### A practical introduction to Fourier Analysis of speech signals. Part I: Background

A practical introduction to Fourier Analysis of speech signals. Part I: Background Let's assume we have used supersine.m to synthesize with just 3 sine components a rough approximation to a square wave,

### Final Report. Hybrid Synth by Synthesize Me. Members: Jamie Madaris Jack Nanney

Final Report Hybrid Synth by Synthesize Me Members: Jamie Madaris Jack Nanney Project Abstract Our Senior Design project is a Digitally-Controlled Analog Synthesizer. The synthesizer consists of both digital

### Chapter 3: Sound waves. Sound waves

Sound waves 1. Sound travels at 340 m/s in air and 1500 m/s in water. A sound of 256 Hz is made under water. In the air, A) the frequency remains the same but the wavelength is shorter. B) the frequency

### T = 1 f. Phase. Measure of relative position in time within a single period of a signal For a periodic signal f(t), phase is fractional part t p

Data Transmission Concepts and terminology Transmission terminology Transmission from transmitter to receiver goes over some transmission medium using electromagnetic waves Guided media. Waves are guided

### ANALYZER BASICS WHAT IS AN FFT SPECTRUM ANALYZER? 2-1

WHAT IS AN FFT SPECTRUM ANALYZER? ANALYZER BASICS The SR760 FFT Spectrum Analyzer takes a time varying input signal, like you would see on an oscilloscope trace, and computes its frequency spectrum. Fourier's

### Chapter 21. Superposition

Chapter 21. Superposition The combination of two or more waves is called a superposition of waves. Applications of superposition range from musical instruments to the colors of an oil film to lasers. Chapter

### Frequency Response of FIR Filters

Frequency Response of FIR Filters Chapter 6 This chapter continues the study of FIR filters from Chapter 5, but the emphasis is frequency response, which relates to how the filter responds to an input

### Topic 4: Continuous-Time Fourier Transform (CTFT)

ELEC264: Signals And Systems Topic 4: Continuous-Time Fourier Transform (CTFT) Aishy Amer Concordia University Electrical and Computer Engineering o Introduction to Fourier Transform o Fourier transform

### Chapter4: Superposition and Interference

Chapter4: Superposition and Interference Sections Superposition Principle Superposition of Sinusoidal Waves Interference of Sound Waves Standing Waves Beats: Interference in Time Nonsinusoidal Wave Patterns

### ME231 Measurements Laboratory Spring Fourier Series. Edmundo Corona c

ME23 Measurements Laboratory Spring 23 Fourier Series Edmundo Corona c If you listen to music you may have noticed that you can tell what instruments are used in a given song or symphony. In some cases,

### B3. Short Time Fourier Transform (STFT)

B3. Short Time Fourier Transform (STFT) Objectives: Understand the concept of a time varying frequency spectrum and the spectrogram Understand the effect of different windows on the spectrogram; Understand

### SOFTWARE FOR GENERATION OF SPECTRUM COMPATIBLE TIME HISTORY

3 th World Conference on Earthquake Engineering Vancouver, B.C., Canada August -6, 24 Paper No. 296 SOFTWARE FOR GENERATION OF SPECTRUM COMPATIBLE TIME HISTORY ASHOK KUMAR SUMMARY One of the important

### Chapter 3 Discrete-Time Fourier Series. by the French mathematician Jean Baptiste Joseph Fourier in the early 1800 s. The

Chapter 3 Discrete-Time Fourier Series 3.1 Introduction The Fourier series and Fourier transforms are mathematical correlations between the time and frequency domains. They are the result of the heat-transfer

### Lab 2: Fourier Analysis

Lab : Fourier Analysis 1 Introduction (a) Refer to Appendix D for photos of the apparatus Joseph Fourier (1768-1830) was one of the French scientists during the time of Napoleon who raised French science

### MIDI: sound control Juan P Bello

MIDI: sound control Juan P Bello Interpreting MIDI MIDI code is the language used to communicate between devices Still, each device might decide to interpret the code in its own way according to its capabilities

### A-145 LFO. 1. Introduction. doepfer System A - 100 LFO A-145

doepfer System A - 100 FO A-145 1. Introduction A-145 FO Module A-145 (FO) is a low frequency oscillator, which produces cyclical control voltages in a very wide range of frequencies. Five waveforms are

### 9: FILTERS, ANALYSIS OF PERIODIC SOUNDS

9: FILTERS, ANALYSIS OF PERIODIC SOUNDS A. Introduction Some weeks you synthesized (put together) complex periodic signals with harmonic components. It is also possible to analyze (take apart) a complex

### ENGR 210 Lab 11 Frequency Response of Passive RC Filters

ENGR 210 Lab 11 Response of Passive RC Filters The objective of this lab is to introduce you to the frequency-dependent nature of the impedance of a capacitor and the impact of that frequency dependence

### Speech Signal Processing Handout 4 Spectral Analysis & the DFT

Speech Signal Processing Handout 4 Spectral Analysis & the DFT Signal: N samples Spectrum: DFT Result: N complex values Frequency resolution = 1 (Hz) NT Magnitude 1 Symmetric about Hz 2T frequency (Hz)

### 1/26/2016. Chapter 21 Superposition. Chapter 21 Preview. Chapter 21 Preview

Chapter 21 Superposition Chapter Goal: To understand and use the idea of superposition. Slide 21-2 Chapter 21 Preview Slide 21-3 Chapter 21 Preview Slide 21-4 1 Chapter 21 Preview Slide 21-5 Chapter 21

### BPSK - BINARY PHASE SHIFT KEYING

BPSK - BINARY PHASE SHIFT KEYING PREPARATION... 70 generation of BPSK... 70 bandlimiting... 71 BPSK demodulation... 72 phase ambiguity...72 EXPERIMENT... 73 the BPSK generator... 73 BPSK demodulator...

### ADC Familiarization. Dmitry Teytelman and Dan Van Winkle. Student name:

Dmitry Teytelman and Dan Van Winkle Student name: RF and Digital Signal Processing US Particle Accelerator School 15 19 June, 2009 June 16, 2009 Contents 1 Introduction 2 2 Exercises 2 2.1 Measuring 48

### Echolocation and hearing in bats

Echolocation and hearing in bats Sound transmission Sound properties Attenuation Echolocation Decoding information from echoes Alternative calling strategies Adaptations for hearing in bats Websites How

### Experiment 16 ANALOG FOURIER ANALYSIS. Experiment setup 4. Prelab problems 6. Experiment procedure 7. Appendix A: an effective high-q filter A-1

16-i Experiment 16 ANALOG FOURIER ANALYI Introduction 1 Theory 1 Experiment setup 4 Prelab problems 6 Experiment procedure 7 Appendix A: an effective high-q filter A-1 16-ii 16-1 1/31/14 INTRODUCTION In

Performance of an IF sampling ADC in receiver applications David Buchanan Staff Applications Engineer Analog Devices, Inc. Introduction The concept of direct intermediate frequency (IF) sampling is not

### Analog Measurand: Time Dependant Characteristics

Chapter 4 Analog Measurand: Time Dependant Characteristics 4.1 Introduction A parameter common to all of measurement is time: All measurands have time-related characteristics. As time progresses, the magnitude

### Constructive and Destructive Interference Conceptual Question

Chapter 16 - solutions Constructive and Destructive Interference Conceptual Question Description: Conceptual question on whether constructive or destructive interference occurs at various points between

### EE 179 April 28, 2014 Digital and Analog Communication Systems Handout #21 Homework #3 Solutions

EE 179 April 28, 214 Digital and Analog Communication Systems Handout #21 Homework #3 Solutions 1. DSB-SC modulator (Lathi& Ding 4.2-3). You are asked to design a DSB-SC modulator to generate a modulated

### Chapter 18 4/14/11. Superposition Principle. Superposition and Interference. Superposition Example. Superposition and Standing Waves

Superposition Principle Chapter 18 Superposition and Standing Waves If two or more traveling waves are moving through a medium, the resultant value of the wave function at any point is the algebraic sum

### Physics 53. Oscillations. You've got to be very careful if you don't know where you're going, because you might not get there.

Physics 53 Oscillations You've got to be very careful if you don't know where you're going, because you might not get there. Yogi Berra Overview Many natural phenomena exhibit motion in which particles

### Analog and Digital Signals, Time and Frequency Representation of Signals

1 Analog and Digital Signals, Time and Frequency Representation of Signals Required reading: Garcia 3.1, 3.2 CSE 3213, Fall 2010 Instructor: N. Vlajic 2 Data vs. Signal Analog vs. Digital Analog Signals

### Introduction to acoustic phonetics

Introduction to acoustic phonetics Dr. Christian DiCanio cdicanio@buffalo.edu University at Buffalo 10/8/15 DiCanio (UB) Acoustics 10/8/15 1 / 28 Pressure & Waves Waves Sound waves are fluctuations in

### Lecture 13: More on Perception of Loudness

Lecture 13: More on Perception of Loudness We have now seen that perception of loudness is not linear in how loud a sound is, but scales roughly as a factor of 2 in perception for a factor of 10 in intensity.

### Fourier Analysis. Evan Sheridan, Chris Kervick, Tom Power Novemeber

Fourier Analysis Evan Sheridan, Chris Kervick, Tom Power 11367741 Novemeber 19 01 Abstract Various properties of the Fourier Transform are investigated using the Cassy Lab software, a microphone, electrical

### Signals and Systems. Chapter SS-8 Communication Systems. ShoushuiWei. Spring SDU-BME Sep08 Dec08. Introduction

Spring 2012 Signals and Systems Chapter SS-8 Communication Systems ShoushuiWei SDU-BME Sep08 Dec08 Figures and images used in these lecture notes are adopted from Signals & Systems by Alan V. Oppenheim

### f x a 0 n 1 a 0 a 1 cos x a 2 cos 2x a 3 cos 3x b 1 sin x b 2 sin 2x b 3 sin 3x a n cos nx b n sin nx n 1 f x dx y

Fourier Series When the French mathematician Joseph Fourier (768 83) was tring to solve a problem in heat conduction, he needed to epress a function f as an infinite series of sine and cosine functions:

### What is a Filter? Output Signal. Input Signal Amplitude. Frequency. Low Pass Filter

What is a Filter? Input Signal Amplitude Output Signal Frequency Time Sequence Low Pass Filter Time Sequence What is a Filter Input Signal Amplitude Output Signal Frequency Signal Noise Signal Noise Frequency

### Implementation of Digital Signal Processing: Some Background on GFSK Modulation

Implementation of Digital Signal Processing: Some Background on GFSK Modulation Sabih H. Gerez University of Twente, Department of Electrical Engineering s.h.gerez@utwente.nl Version 4 (February 7, 2013)

### Audio Engineering Society. Convention Paper. Presented at the 129th Convention 2010 November 4 7 San Francisco, CA, USA

Audio Engineering Society Convention Paper Presented at the 129th Convention 2010 November 4 7 San Francisco, CA, USA The papers at this Convention have been selected on the basis of a submitted abstract

### Part 2: Receiver and Demodulator

University of Pennsylvania Department of Electrical and Systems Engineering ESE06: Electrical Circuits and Systems II Lab Amplitude Modulated Radio Frequency Transmission System Mini-Project Part : Receiver

### Teaching Fourier Analysis and Wave Physics with the Bass Guitar

Teaching Fourier Analysis and Wave Physics with the Bass Guitar Michael Courtney Department of Chemistry and Physics, Western Carolina University Norm Althausen Lorain County Community College This article

### ANALYTICAL METHODS FOR ENGINEERS

UNIT 1: Unit code: QCF Level: 4 Credit value: 15 ANALYTICAL METHODS FOR ENGINEERS A/601/1401 OUTCOME - TRIGONOMETRIC METHODS TUTORIAL 1 SINUSOIDAL FUNCTION Be able to analyse and model engineering situations

### Chapter 1. Oscillations. Oscillations

Oscillations 1. A mass m hanging on a spring with a spring constant k has simple harmonic motion with a period T. If the mass is doubled to 2m, the period of oscillation A) increases by a factor of 2.

### Modulation methods. S-72. 333 Physical layer methods in wireless communication systems. Sylvain Ranvier / Radio Laboratory / TKK 16 November 2004

Modulation methods S-72. 333 Physical layer methods in wireless communication systems Sylvain Ranvier / Radio Laboratory / TKK 16 November 2004 sylvain.ranvier@hut.fi SMARAD / Radio Laboratory 1 Line out

### The Sonometer The Resonant String and Timbre Change after plucking

The Sonometer The Resonant String and Timbre Change after plucking EQUIPMENT Pasco sonometers (pick up 5 from teaching lab) and 5 kits to go with them BK Precision function generators and Tenma oscilloscopes