EE513Audio Signals and Systems

Noise

Kevin D. DonohueElectrical and Computer Engineering

University of Kentucky

Quantization NoiseSignal amplitudes take on a continuum of values. A discrete signal must be digitized (mapped to a finite set of values) to be stored and processed on a computer/DSP

Digital SignalDiscrete-time Signal

QuantizerAnalog Signal

Coder

11 10 01 00

)(nTxa )(ˆ nTx

)(ˆ nx

Quantization Error and Noise

Quantization has the same effects as adding noise to the signal as long as the rounding error is small compare to the original signal amplitude:

11 10 01 00

)(ˆ)()( nTxnTxnT aq )(ˆ)()( nTxnTnTx qa

)(txa )(nTxa )(ˆ nTxAnalog Discrete Digital

Intervals between quantization levels are proportional to the resulting quantization noise since they limit the maximum rounding or truncation error.

For uniform quantization, the quantization level interval is the maximum signal range divided by the number of quantization intervals.

Quantization NoiseOriginal CD clip quantized at 16 bits (blue)

Quantized at 6 bits (red)

Quantized at 3 bits (black)

101

102

103

104

105

-80

-60

-40

-20

0

20

Hertz

dB

PSDs of Quantized Signal; Song -Tell Me Ma

16 bit

3 bit

6 bit

Quantization Noise Analysis

Assume is a uniformly distributed (amplitude), white, stationary process that is uncorrelated with the signal.

• Show that the signal to quantization noise ratio (SNRq) for a full scale range (FSR) sinusoid, quantized with B bit words is approximately:

• Note this is the SNR for a signal amplitude at FSR, signals with smaller amplitudes. What would be the formula for a sinusoid with an X% FSR?

)(nq

dB 8.16SNRq B

Homework 4.1

• Derive a formula for SNRq similar to the one on last slide (in dB) for a sinusoid that is X% of the FSR in amplitude.

Noise generated from a source inside a room will undergo frequency dependent propagation, absorption and refection before reaching the sink. Thus, the room effectively filters the sound.

Sound impinging on surfaces in the room will be absorbed, reflected, or diffused.

Room Noise

Heat

DirectSound

Absorption

DirectSoun

d

Reflection

SpecularReflected

Sound

DirectSoun

d

Diffusion

DiffuseScattered

Sound

Transmission

Reflected and reverberant sounds become particularly bad distractions because they are highly correlated with the original sound source. The use of absorbers and diffusers on reflective surfaces can cut down the reverberation effects in rooms.

The model for a signal received at a point in space from many reflections is given as:

where n(t) denotes the attenuation of each reflected signal due to propagation through the air and absorption at each reflected interface and n is the time delay associated with the travel path from the source to the receiver. The signal in the frequency domain is given by:

Reflection Absorption Effects

N

nnn dtstr

1 0

))(()()(

N

nnn fjffSfR

1

)2exp()()()(

Reverberant Sound Travel

LS

EF1

EF2

EF3

EF4

RF2

RF1

RF3

D

The near or direct field (D)The free or early field (EF1 and EF2)The reverberant or diffuse field (RF1 to RF3)

Decay of Reverberant Sound Field

Time

Sou

nd L

evel

Direct Sound

Reverberation

Initial Time Delay Gap

60 dB

Reverberation Time

The time it takes for the reverberant sound field to decayby 60dB has become a standard way to characterize reverberation in room acoustics.

For a space with many randomly distributed reflectors (typically large rooms) reverberation time (RT60 ) is defined as the amount of time for the sound pressure in a room to decrease by 60 dB from its maximum. The time is statistically predicted from the room features with the Sabine equation:

where V is the volume of the room in cubic meters Si is the surface area of the ith surface in room (in square meters) ai is the absorption coefficient of ith surface m is the absorption coefficient of air.

Discuss: The relationship between absorption, volume, and RT.

Room Reverberation Time

VfmfaS

VfRT

N

i

ii )(4)(

161.)(

1

60

Room Response to White Noise Input

Data collected and spectrogram computed by H.L. FournierNote frequency dependence on of decay time.

ExampleGiven the simulated reverb signal compute the RT60. Find the autocorrelation function and try to estimate the delays associated with the major scatterers.

% Create reverb signal

[y,fs] = wavread('clap.wav'); % Read in Clap sound

% Apply simulated reverb signal

yout1 = mrevera(y,fs,[30 44 121]*1e-3,[.6 .8 .6]);

taxis = [0:length(yout1)-1]/fs;

% Compute envelope of signal

env = abs(hilbert(yout1));

figure(1)

plot(taxis,20*log10(env+eps)) % Plot Power over time

hold on

% Create Line at 60 dB below max point and look for intersection point

mp = max(20*log10(env+eps));

mp = mp(1);

dt = mp-60;

plot(taxis,dt*ones(size(taxis)),'r'); hold off; xlabel('Seconds')

ylabel('dB'); title('Envelope of Room Impulse Response')

% Compute autocorrelation function of envelop and look for peaks % to indicate delay of major echoes

maxlag = fix(fs*.5);

[ac, lags] = xcorr(env-mean(env), maxlag);

figure(2)

plot(lags/fs,ac)

xlabel('seconds')

ylabel('AC coefficient')

% Compute autocorrelation function of raw and look for peaks to

% indicate delay of major echoes

[ac, lags] = xcorr(yout1, maxlag);

figure(3)

plot(lags/fs,ac)

xlabel('seconds')

ylabel('AC coefficient')

Room ModesThe air in a (small) rectangular room has natural modes of vibration given by:

where c is the speed of sound in the room p, h, and r are integers 0,1,2, …., and L, W, and H are the length, width, and height of the room.

222

2

H

r

W

q

L

pcf

Efficiency – Output power over Input power (including that of the power supply). Distortion – Total harmonic distortion (THD). For a sinusoidal signal input, THD is the ratio of power at all harmonic frequencies Pi (excluding the fundamental P1) to the power at the fundamental frequency.

where PT is total signal power

Fidelity – Flatness of frequency response characterized by frequency range and transfer function variation in that range.

Amplifiers and Distortion

1

1

1

2

P

PP

P

PTHD Ti

i

Given the transfer characteristic for a class B amplifier below, compute the THD for a 3 volt input sinusoid.

Example

Vin

Vout

0.6v

-0.6v

7v

3v

-7v

-3v

Class A - Low distortion, bad efficiency. Output stage with single transistor requires DC biased output (10-20% efficiency).

Class B - Crossover distortion, good efficiency. Output stage has 2 transistors so bias current is zero (~80% efficient).

Class AB – Reduced crossover distortion, good efficiency. Output stage has 2 transistors with biasing to push signal out of crossover distortion range.

Class D – Moderate distortion, high efficiency, operates in switch mode. Good for battery driven applications.

Amplifier Classes

Center Clip Distortion

0 0.005 0.01 0.015 0.02 0.025 0.03 0.035 0.04-3

-2

-1

0

1

2

3

seconds

am

plit

ud

e

OriginalDistorted

0 500 1000 1500 2000 2500 3000 3500 4000-100

-80

-60

-40

-20

0

Hz

dB

OriginalDistorted

fo = 200 HzTHD = 4.13%

Harmonic Peak Heights = [-8, -23, -29, -37, -47, -55, -47, -46, -49, -57];

10/8

10/5710/3710/2910/23

10

10101010

THD

Given the transfer characteristic for a class AB amplifier below, compute the THD for a 3 volt input sinusoid.

Example

Vin

Vout7v

3v

-7v

-3v

1.75v-1.75v

Clip/Overload Distortion

0 500 1000 1500 2000 2500 3000 3500 4000-100

-80

-60

-40

-20

0

Hz

dB

OriginalDistorted

0 0.005 0.01 0.015 0.02 0.025 0.03 0.035 0.04-3

-2

-1

0

1

2

3

seconds

am

plit

ud

e

OriginalDistorted

fo = 200 HzTHD = 4.14%

Harmonic Peak Heights = [-7, -21, -46, -37, -44, -49, -45, -72, -49, -55];

10/7

10/5510/3710/4610/21

10

10101010

THD