3 signal processing essentials representation of audio signals

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Signal Processing Essentials

Representation of Audio Signals

06.10.2011, 52.219, Waldo Nogueira


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Group My Self

Contact Details Email: [email protected]

Telephone: +34 93 542 2806Web: https://dtic.upf.edu/~wnogueira (underconstruction)

Office: 55.314 Address: Music Technology Group

Tnger Building - CommunicationCampus-PoblenouUniversitat Pompeu FabraRoc Boronat, 13808018 Barcelona

Moodle:


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GroupContents

Review of the last 3 Lectures: From Analogue to Digital


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GroupLecture 1: Introduction

medi, dades, codis i artefactes la finestra daudici. Visi histrica de la

codificaci digital teorema de Nyquist: mostreig i aliasing caracterstiques dels filtres analgics anti-aliasing

i reconstructor freqncies de mostreig estndard (32/44.1/48

KHz)mostreig i retenci (sample&hold) en A/D importncia del rellotge de mostreig (clock):jitteri PLL efecte dapertura (en A/D i en reconstrucci D/A)


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GroupLecture 2: Quantization

quantitzaci error de quantitzaci (distorsi harmnica)

dither: definici i tcniques (rectangular,triangular, gaussi) aplicacions del ditheri criteris de selecci en

lentorn professional tecnologia bsica dimplementaci de la

conversi A/D/A


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GroupLecture 3: Sampling, PCM and more

codificaci no-PCM: diferencial (delta D, DPCM) altres codificacions PCM: sigma-delta -D, -

DPCM sobremostreig (oversampling): definici, teoria

de la informaci noise-shaping (NS): definici i exemples

dimplementaci diagrama global de la conversi A/D/A (amb

ovesampling i NS) conversors comercials en lestudi

denregistrament: parmetres i criteris relaci entre formats comercials i tipus de

conversi (CD, DVD-A, SACD, estat slid)


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GroupContents for Today: Part I/II

After Analogue to Digital conversion Part I: Introduction and Overview:

Source Coding in a digital information system

Quantification and Coding of signals with PCM Problem: Large amount of information

PCM for Audio and Speech Signals

Information Theory: The message

Overview: Structure of the next 6 Lectures

Part II: Signal Processing Fundamentals

Spectra of Analogue Signals, Convolution and Filtering, UniformSampling

Discrete time signal processing (Transforms, DFT, DCT, STFT)

Differential equations and digital filters (Poles,Zeros,Freq. Resp.

Review of MultiRate signal processing (Down/Up-samping, QMF)

Discrete-Time Random Signals Information Theory


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GroupContents

PART I


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GroupDigital Coding

Representation of Analog signals using binarydigital sequences

Goals: Acceptable quality with the minimum number of bits

For a fixed speed to obtain the minimum degradation


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Source coding in digitalinformation system

SourceCoder

ChannelCoder Modulator

Channel

De-Modulator

ChannelDe-coder

SourceDe-coder

BinarySymbols

Source

Sink

Noise+

Interference

Discrete Channel


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Quantification and Coding of Signalswith PCM

Time


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GroupProblem

The digital representation of Audio Signalsrequires relatively large data rates

Digital Audio Signal = 24 x Digital Speech Signal

2 x 768 kbit/s 64 kbit/s


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Group

PCM-Format for Audio and SpeechSignals


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GroupAdvantages of digital coding

Regeneration Possibility to clean pulse. When reconstructing the signal

it is possible to improve the SNR

In analog comunications if signal is amplified, the noise isamplified too

Error protection

Encripting StorageMultiplexing (time and frequency) Less degradationUse of signal processing techniques


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GroupBit Rate

At Source Coder output we obtain:

Bits/sample: R

Sampling Rate: Fs

Transmission speed I: FsR bits/second

Quantization

&Sampling

Source

Coder

Source

Fs: samples/sR: bits/sample

How can be the transmission rate be reduced, or usedo timall Information Theor


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GroupInformation Theory: Message

R H

Redundant

(known)

Not Redundant

(unknown)

Irrelevant

(nomeaning

)

Releva

nt

(hasmea

ning)

Interesting


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General Block Diagram of an Audiocoder

Time-FrequencyTransform

Quantizationand Coding

FrameConstruction

Psychoacoustic Model

PCM- Signal

706 KBIT/SQuantizationand Coding


Coded AudioSignal

128 KBIT/S


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GroupEvaluation of coding schemes

4 parameters to evaluate a coding scheme 1) Speed

64 Kbps 2,4 Kbps

256 Kbps 16 Kbps

1410 Kbps 128 Kbps

Quality

Objective Measures Subjective Measures

Complexity

MIPS Delay

< 100 ms


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GroupOverview

Theory 1,2,3,4.5 (until slide 27): Done by Enric Gine Theory 4.5: Signal Processing Essentials

Theory 5: Quantization and Entropy Coding

(coming from sigma-delta)

Theory 6: Time-Frequency Analysis

Theory 7: Psychoacoustics & Psycho.Models

Theory 8: Bit Allocations Strategies

Theory 9: Linear Prediction and Narrow/Wideband coding

Seminar 1, 2, 3: Done by Enric Gine

Seminar 4: Audio Coding Standards 1 (CELP, AC-3, mp-1-2-3,

AAC) Seminar 5: Audio Coding Standards 2 (mpeg-Surround, USAC)

Seminar 6: Losless Audio Coding

Seminar 7: Quality Measures for Audio Coding (First Seminar?)

Callibration of the system from input/output(Look at Watkinson)

Seminar 8: Sinusoidal Coders


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GroupOverview: Labs

Lab 1:Enric Lab 2: Signal Processing Essentials Lab 3: Quantization & Entropic Coding Lab 4: Psychoacousitcs Lab 5: To be defined!!!


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General Block Diagram of an Audiocoder

Time-FrequencyTransform


FrameConstruction

Psychoacoustic Model

PCM- Signal

706 KBIT/SQuantizationand Coding


Coded AudioSignal

128 KBIT/S

Theory 5

Theory 6

Theory 7

Theory 8


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GroupContents

PART II


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GroupContents

PART II: Introduction Spectra of Analog Signals Review of convolution and filtering Uniform Sampling

Discrete-time signal processing Transforms for Discrete-Time Signals The Discrete and the Fast Fourier Transform The Discrete Cosine Transform The Short-Time Fourier Transform

Difference equations and digital filters

The transfer and the frequency response functions Poles, Zeros, and Frequency Response Examples of digital filters for audio applications

Review of Multirate signal processing Down-sampling by an integer Up-sampling by an integer Sampling rate changes by Noninteger factors Quadrature mirror filter banks

Discrete-Time Random Signals Random signals processed by LTI Digital Filters Autocorrelation Estimation from Finite-Length Data

Review on Information Theory Entropy, Shannon, Rate Distortion


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GroupSpectra of Analoge Signals (1)

Frequency Spectrum of an analog signal isdescribed in terms of the continuous Fourier

transform (CFT). ,

Where wis the frequency in radians per second (rad/s), w = 2f,where fis the frequency in Hz.

Example 1: The pulse-sinc CFT pair


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GroupSpectra of Analog Signals (1)

Example 2:

Loss of resolution andspectral leackage (timedomain truncation)

M i


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GroupSpectra of Analog Signals

The inverse CFT is given by

12

In CFT theory, x(t) and X(w) are called atransform pair, i.e.,

x(t) X(w)

M i


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GroupSpectra of Analog Signals

In real life signal processing, all signals have finite length, Time domain truncation always occurs.

Truncation done using rectangular windows.

To smooth out frame transitions and control spectral leakage(sidelobes), the signal is often tappered prior to truncation usingwindows such as Hamming, Barlett, trapezoidal windows.

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GroupReview of Convolution and Filtering

A linear time-invariant (LTI) system

A linear filter satisfies the property ofsuperposition: y(t), is the convolution of the input, x(t),

with the filter impulse response, h(t).

Convolution is represented by the integral

.

LTI system

h(t)

Y(t) = x(t)*h(t)x(t)

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GroupReview of Convolution and Filtering

The CFT of the impulsive response, h(t), is thefrequency response of te filter, i.e.,

h(t) H(w). Example: First order first-order RC circuit (low

pass filter)

Impulse response is a decaying exponential, and frequencyresponse is a rational function. This function is complex-valuedand its magnitude represents the gain of the filter with respectto frequency at steady state. A sinusoid is scaled in amplitude

and phase is shifted with respect to phase response

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GroupUniform Sampling

Nyquist Fs 2B

Multiplication in timedomain is convolution infrequency domain.

Sampling is

multiplicationspectrum isconvolution of spectra

with pulse train: 1 .

An ideal LPF can recover thebaseband and hence perfectreconstruct the analog signal fromthe digital

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GroupUniform Sampling

Reconstruction process:

The reconstruction LPFinterpolates between thesamples and reproducesthe analog signal

The interpolation becomesevident once the filtering

operation is interpreted intime domain asconvolution

Reconstruction occurs by

interpolating with the sincfunction, which is theimpulse response of theideal LPF.

Note that if Fs is less than 2B, aliasing occurs, andperfect reconstruction is no longer possible

In real life, the analog signal is not ideally badlimited ann the sampling process is not perfect(sampling pulses have finite amplitude and duration) Aliasing

To reduce aliasing, the signals is prefiltered by ananti-aliasing low pass and usually over-sampled

(ws>2B) sigma delta

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Group2.5 Discrete-time signal processing

Audio coding algorithms operate on a quantizeddiscrete-time signal.

Prior to compression, most algorithms requirethat the audio signal is acquired with high fidelitycharacteristics.

Typically it is assumed that the signal is band-limited at 20 kHz, sampled at 44.1 kHz and

quantized at 16 bits per sample

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Group2.5.1 Transforms for Discrete-Time signals

Discrete-time signals are described in thetransform domain using thez-transform and thediscrete-time Fourier transform (DTFT).

The z-transform is defined as: , where z is complex If the z-transform is evaluated on the unit circle,

i.e, forz , 2

Then the z-transform becomes the discrete timeFourier transform (DTFT)

.

Music 2 5 2 The Discrete and the Fast Fourier


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2.5.2 The Discrete and the Fast FourierTransform

The Fourier transform is developed by startingfrom the DTFT analysis expression, andconsidering a finite length signal consisting ofNpoints, i.e.,

Furthermore, the frequency-domain signal issampled uniformly at N points within one period, = 0 to 2, i.e.,

k =

, K= 0,1,...,N-1

It is typical in the DSP literature to replace kwith the frequency index kand hence, it can bewritten as:

.

/ , k = 0,1,2,...,N-1



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2.5.2 The Discrete and the Fast FourierTransform (2)

The sampling in the frequency domain forcesperiodicity in the time domain, i.e.,

x(n) = x(n+N).

We also have periodicity in the frequencydomain, X(k) = X(k+N), because the signal in

the time domain is also discrete. Theseperiodicities create circular effects whenconvolution is performed by frequency-domain

multiplication, i.e,x(n) h(n) X(k)H(k),

Where x(n) h(n) = .



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2.5.2 The Discrete and the Fast FourierTransform (2)

The N-point inverse DFT (IDFT) is written as:

1

, 0,1, , 1

The DFT can be computed efficiently using the fastfourier transform (FFT).

FFT takes advantage of redundancies in the DFT sumby decimating the sequence into subsequences witheven and odd indices.

It can be shown that ifNis a radix-2 integer, the N-point DFT can be computed using a series ofbutterfly stages.

The complexity associated with the DFT is of theorderN2and for the FFT is roughly of the order of

Nlog2N

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2 5 3 Th Di t C i T f


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Group2.5.3 The Discrete Cosine Transform

The discrete cosine transform (DCT) of x(n) canbe defined as

, 0 1,Where c(0) = 1/ 2, and c(k) = 1 for 1kN-1.

Depending on the periodicity and symmetry of theinput signal, x(n), the DCT can be computed using

different orthonormal transforms (usually DCT-1,DCT-2, DCT-3, and DCT-4) (More details will begiven on Chapter 6)

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h l 2 5 4 Th Sh t Ti F i T f


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Group2.5.4 The Short-Time Fourier Transform

Spectral analysis of non-stationary signals cannot be accommodatedby classical Fourier transform since the signal has time-varyingcharacteristics

Time-varying spectral analysis can be performed using the short-timeFourier transform (STFT).

,

,

W here = wT = 2fT is the normalized frequency in radians, and h(n)is the sliding analysis window The synthesis expression (inverse transform) is given by

12 ,

If n = m nad h(0) = 1, then x(n) can be obtained from previous Eq. The basic assumption in this type of analysis-synthesis is that the

signal is slowly time-varying and can be modeled by its short-timespectrum.

The temporal and spectral resolution of the SFTF are controlled by thelength of the window (usually constrained to be about 5-20 ms andhence spectral resolution is sacrificed)

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T h l 2 5 4 Th Sh t Ti F i T f


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Group2.5.4 The Short-Time Fourier Transform

The sequence, h(n), can also be viewed as theimpulse response of a LTI filter, which is excitedby a frequency-shifted signal. This leads to the

filter-bank interpretation of the STFT, i.e, for adiscrete frequency variable k = k(), k =0,1,,N-1 and and N chosen such that thespeech band is covered.

Then the analysis expression is written as: ,

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T h l g2.5.4 The Short-Time Fourier


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Group Transform

And the synthesis expression is:

, ,

Where is the signal reconstructed withinthe band of interest. If h(n), , and N are chosencarefully, the reconstruction given in previous slideis depicted in the following Figure, where

.

The k-th channel of the analysis-synthesis filterbank

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Technology Overlap Add Decomposition


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GroupOverlap-Add Decomposition

Consider breaking an inputsignal x into frames using afinite, zero-phase, length Mwindow w. Then we may

express the mth windowed dataframe as:

, ,

where,

R frame step (hop size)m frame index

The hop size is the number ofsamples between the begin-

times of adjacent frames.Specifically, it is the number ofsamples by which we advanceeach successive window

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Technology Overlap Add Decomposition


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GroupOverlap-Add Decomposition

For frame-by-frame spectral processing to work,we must be able to reconstruct x from theindividual overlapping frames, ideally by simply

summing them in their original time position

= Hence, x = 1, ,This is the constant-overlap-add (COLA) constraintfor the FFT analysis window. It has also been calledpartition of unity transform

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Technology 2 6 Difference Equations and Digital Filters


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Group2.6 Difference Equations and Digital Filters

Digital filters are characterized by differenceequations of the form

y(n) is given as a linear combination of present and past inputsminus linear combination of past outputs (feedback term).

The parameters ai and bi are the filter coefficients and controllfrequency response characteristics of a digital filter.

Filter coefficients can be made adaptive (time-varying) IIR filter (when feedback coefficients are non zero, frequency

response is infinitely long).

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Group Filters

The impulse response h(n) is:

In statistical signal representation (ARMA)The filter coefficients are chosen such that the filter is stable

An input-output equation of a causal filter can also bewritten in terms of the impulse response of the filter, i.e.,

.

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Technology2.7 The Transfer and the Frequency


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Group Response Functions

The z-transform of the impulse response of afilter is called the transfer function is given by

.

Considering the difference equation, we can alsoobtain the transfer function in terms of filterparameters, i.e.,

1 The ratio of output input in the z domain gives thetransfer function in terms of the coefficients

0 0

1 1

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The frequency response function is a special caseof the transfer function of the filter. That is for , then

2.7.1 Poles, Zeros and Frequency Response

A z domain function, H(z), can be written in terms of its

poles and zeros as follows:

Where are the zeros and poles of H(z) and G is cte

The location of poles and zeos affect shape of freq response

Magnitude of freq.response can be writte as:

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It is evident that when anIsolated Zero is close to the

it unit circle, it will give rise to a peak in the magnitudefrequency response will assume a small value at thatfrequency

When an isolated pole is close to unit circle it will give riseto a peak in the magnitude frequency response at that

frequency. In speech processing,

the presence of poles in z domain representations of the vocal tract,has been associated with the speech formants.

Formant synthesizers use the pole locations to form synthesis filters forcertain phonemes.

The presence of zeros has been associated with the coupling of nasaltract. For example, zeros associate with nasal such as m and n.

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Technology2.8 Review of multirate signal

i (MSP)


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Group processing (MSP)

MSP involves the change of sampling rate whilethe signal is in the digital domain Applications: Reduce algorithmic (SW/HW) complexity

Increase resolution

Oversampling A/D (less requirem. for antialiasing filter),but causes data-rate increase Downsampling required

Downsampling requires anti-aliasing filter low-pass indigital domain

Sigma delta changes complexity from analog domain into

digital domain

For D/A conversion signal is interpolated in the digitaldomain reducing requirements on the analog

reconstruction (interpolation) filter

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Technology 2.8.1 Downsampling by an Integer


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Group2.8.1 Downsampling by an Integer

Downsampling: Increase the sampling period/Decreasesampling frequency and data rate of digital signal:

Given the DTFT transform pairs

and

It can be shown that the DTFT of the original and decimatedsignal are related by

1

/

.

Downsampling introduces L copies of the original DTFTthat are both amplitude and frequency sacaled by L May introduce aliasing.

Aliasing can be eliminated if DTFT is bandlimited to /L,

i.e:

0,

.

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Technology 2.8.1 Downsampling by an Integer


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Group8 o sa p g by a tege

Example: DTFTs of the signal during downsampling process

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G2.8.2 Upsampling by an Integer


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Groupp p g y g

Involves reducing the sampling period byintroducing additional regularly spaced samplesin the signal sequence

The DTFT of the up-sampled signal relates to the

DTFT of the original signal as follows: .

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G2.8.2 Upsampling by an Integer


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Groupp p g y g

Therefore, the DTFT of the up-sampled, sdescribed by a series of compressed images ofthe DTFT of the original signal located at integer

mutliples of 2/M rads. To complete the upsampling process, an

interpolation stage is required that fills

appropriate values in the time-domain to replacethe artificial zero-valued samples introduced bythe sampling

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2.8.3 Sampling Rate Changes byNonInteger Factors


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Group NonInteger Factors

Can be accomplished by cascading upsamplingand down-sampling The upsampling stage precedes the downsampling stage

and, The low pass interpolation and antialiasing filters are

combined into one filter whose bandwidth is the minimum

of the two filters

Example: We wand a non integer sampling periodmodification of such that Tnew = 12T/5.

In this case we choose L=12 and M=5.

Hence, the bandwidth of the low-pass filter is the minimum

of /12 and /5.

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GroupQuadrature Mirror Filter Banks


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Analysis of a signal in a Perceptual Audio Coderis accomplished using filter-banks or frequency-domain transformations (or combinations)

The filterbank is used to decompose the signalinto several frequency subbandsDifferent coding strategies are then derived and

imlemented in each subband (subband coding)- Aliasing between subbands

because of imperfect freq.responses- Aliasing effects no perfect

reconstruction

- Solution: Combine up-&downoperations with appropriate filterdesigns

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GroupQuadrature Mirror Filter


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Perfect reconstruction fiitlerbank is calledquadrature mirror fitler bank (QMF)

Analysis H0(z) and H1(z) + downsampling Synthesis F0(z) and F1(z) If the process includes quantizers, those will be

placed after the downsampling stages

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The input signal x(n) is first filtered anddownsampled

The DTFT is

Plot of DTFT of the original and down-sampledsignals is presented here:

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The reconstructed signal , is derived by addingthe contributions from the up-sampling and theinterpolations of the low and the high band.

It can be shown that the reconstructed signal in z-domain has the form

12 0 0 1 1 12 0 0 1 1

The signal X(-z) is associated with the alisaing term.The aliasing term can be cancelled by designingfilters to have the following mirror symmetries:

F0(z) = H1(-z) and F1(z)=-H0(-z)

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Under these conditions, the overall transfer function ofthe filter bank can then be written as

0 0 1 1 . If T(z) = 1, then the filter bank allows perfect

reconstruction.

Perfect delayless reconstruction is not realizable, but anall-pass filter bank with linear phase characteristics can be

designed easily For example a first order FIR filter

0 1 1 1

Results in alias free reconstruction. The overall transferfunction is:

1 - 1 2

The signal is reconstructed within a delay of one sampleand with an overall gain of 2

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QMF filter banks can be cascaded to form treestructures.

If we represent the analysis stage of a filter bank

as a block that divides the signal in low and highfrequency subbands, then by cascading severalof such blocks, we can divide the signal into

smaller subbands (association with wavelettransform theory)

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GroupDiscrete-Time Random Signals


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Signals can be classified as deterministic or random Deterministic: Values at any point in time can be defined

precisely by a math. Equation (e.g. x(n)=sin(n/4))

Random: Uncertain values and are usually described usingstatistics

A discrete-time random process involves an ensemble ofsequences x(n,m) where m is the index of the m-th

sequence in the ensemble and n is the time index In practice one does not have access to all possible sample

signals of a random process. Therefore, the determination ofthe statistical structure of a random process is often done from

the observed waveform. This approach becomes valid andsimplifies if the random signal is Ergodic.

Ergodicity: Statistics of a random process can bedetermined using time-averaging operations on asingle observed signal

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Ergodicity requires that statistics of the signalare independent of the time of observation

Stationarity: Random process is widesensestationary it its statistics, upto the second order,are independent of time.

Although it is difficult to show analytically thatsignals with various statistical distributions are

ergodic, it can be shown that a stationary zero-mean Gaussian process is ergodic up to secondorder (in many practical applications involving stationarity

process, it is assumed that the process is also ergodic)

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The mean value, x , of the discrete-time, widesense stationary signal, x(n), is a first orderstatistic that is defined as the expected value of

x(n), i.e., lim

12 1 ,

Where E[] denotes statistical expectation The variance is the standard deviation of the

signal. For a zero-mean signal , the variance is

simply .

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The autocorrelation of a signal is a second-orderstatistic defined by:

lim1

2 1 ,

m is autocorrelation lag index. The autocorrelationcan be viewed as a measure of predictability of the

signal in the sense that a future value of a correlatedsignal can be predicted by processing informationassociated with its past values.

For example speech is a correlated waveform, and,

hence, it can be modelled by linear predictionmechanisms that predict its current value from a linearcombination of past values

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Correlation can also be viewed as a measure ofredundancy in the signal: Correlated waveforms can be parameterized in terms of

statistical time-series models; and, hence, represented bya reduced number of information bits

The autocorrelation sequence rxx(m), issymmetric and positive definite, i.e,

0 . Example 1: Autocorrelation of a white noise

signal

,Where is the variance of the noise.Autocorrelation of white noise is an impulse

Uncorrelated signal

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Example 2:

Autocorrelation of the output FIR digital filter, H(z) to awhite noise input of zero mean and unit variance is

2 2 1 3 2 1 2Cross-correlation is a measure of similarity between two

signals. The cross-correlation of signal, x(n), relative to asignal, y(n), is given by

.Similarly, cross correlation of a signal, y(n), relative to a

signal, x(n), is given byr (m) = E[y(n+m)x(n)].

Note that the symmetry property of the cross-correlation isr

(m) = r

(-m).

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The power spectral density (PSD) of a randomsignal is defined as the DTFT of theautocorrelation sequence,

. The PSD is real-valued and positive and

describes how the power of the random process

is distributed across frequency. Example: The PSD of a white noise signal is

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Random Signals Processed by LTIdigital filters


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Review characterization of statistics of the outputof a causal LTI digital filter that is excited byrandom signal.

The output of a causal digital filter can becomputed by convolving the input with itsimpulse response, i.e.,

.

We can derive following expressions for mean,

crosscorrelation, and power spectral density atoutput

| =0

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GroupInformation Theory


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In the paper titled A Mathematical Theory ofCommunication, Claude Shannon wrote:

The fundamental problem of communication is that ofreproducing at one point either exactly or

approximately a message selected at another point

Information Theory (IT): Mathematical framework forapproaching a large class of problems related toencoding, transmission, and decoding information in

a systematic and disciplined way. Since audio (speech, music, etc) is a form of

communication, information theory has served as a

based for audio coding

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Entropy: Can be used to describe the quantity ofinformation 1) The amount of uncertainity before seeing an event

2) The amount of surprise when seeing an event

3) The amount of information after seeing an event

(These three are virtually the same)

According to IT, the information derivable fromoutcome xi depends on its probability P(xi) P(xi ) small large degree of information

P(xi ) largesmall amount of information

The amount of information is defined as:

1

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Interpretation for the logarithm The information for two independent events to occur (where the

joint probability is the multiplication of both individualprobabilities) can be simply carried out by the addition of the

individual information of each event. When the logarithm base is 2, the unit information is called bit.

(1 bit of information is needed to specify the outcome)

X is a discrete random variable taking value xi (symbol)from a finite sample space S={x1,x2,,xi,} (alphabet).Xi is produced from alphabet S according to theprobability distribution of the random variable X.

Entropy of random variable xi

1

H(X) is the amount of information required to specify whatkind of symbol has occurred on average

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GroupSummary


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Concepts covered during this chapter Continuous Fourier Transform

Spectral Leakage effects

Convolution, Sampling, and Aliasing issues

Discrete-time Fourier transform and z-transform

The DFT, FFT, DCT, and STFT basics

Information Theory

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Any Question?

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Thanks!Thanks!

3 signal processing essentials representation of audio signals

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