Noise
Kevin D. Donohue
Electrical and Computer Engineering
University of Kentucky

Signal 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
Analog Signal
Discrete-time
Signal
Quantizer x a
( nT )
Digital Signal x
ˆ
( nT )
Coder x
ˆ
( n )
11
10
01
00

x a
( t )
Analog x a
( nT )
Discrete x
ˆ
( nT )
Digital

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:
 q
( nT )
 x a
( nT )
 x
ˆ
( nT ) x a
( nT )
  q
( nT )
 ˆ
( nT )
 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.
11
10
01
00

Original CD clip quantized at 16 bits (blue)
Quantized at 6 bits (red)
Quantized at 3 bits (black)
PSDs of Quantized Signal; Song -Tell Me Ma
20
0
-20
-40
-60
-80
10
1
10
2
10
3
Hertz
3 bit
16 bit
10
4
6 bit
10
5

 white, stationary process that is uncorrelated with the signal.
•
Show that the signal to quantization noise ratio (SNR q
) for a full scale range (FSR) sinusoid, quantized with B bit words is approximately:
SNR q

6 B

1 .
8 dB
•
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?

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.
Absorption Reflection Diffusion
Heat
Transmission
Direct
Sound
Specular
Reflected
Sound
Direct
Sound
Diffuse
Scattered
Sound
Direct
Sound

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: r ( t )
 n
N 


1 0
 n
(

) s (
 
( t
  n
)) d
 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:
R ( f )

S ( f ) n
N 

1
 n
( f ) exp(
 j 2
 f
 n
)

RF
1
EF
1
S
D
EF
2
EF
3
L RF
2
EF
4
RF
3
The near or direct field (D)
The free or early field (EF1 and EF2)
The reverberant or diffuse field (RF1 to RF3)

Direct Sound
60 dB
Reverberation
Initial Time Delay Gap
Time
Reverberation Time
The time it takes for the reverberant sound field to decay by 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 (RT
60
) 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:
RT
60
( f
V
)

.
161 i
N 

1
S i a i
( f )

4 m ( f ) V where

V is the volume of the room in cubic meters

S i is the surface area of the i th surface in room (in square meters)
 a i is the absorption coefficient of i th
 m is the absorption coefficient of air.
surface
Discuss: The relationship between absorption, volume, and RT.

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

Given 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')

The air in a (small) rectangular room has natural modes of vibration given by: f
 c
2

 p
L


2


 q
W


2


 r
H


2 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.

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
P i
(excluding the fundamental frequency.
P
THD
 i



2
P
1
P i

P
T

P
1
P
1
1
) to the power at the fundamental where P
T is total signal power
Fidelity – Flatness of frequency response characterized by frequency range and transfer function variation in that range.

Given the transfer characteristic for a class B amplifier below, compute the THD for a 3 volt input sinusoid.
V out
7v
-3v
-0.6v
0.6v
-7v
3v
V in

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.

1
2
3
Original
Distorted
0
-1
-2
-3
0 0.005
0.01
0.015
0.02
seconds
0.025
0.03
0.035
Harmonic Peak Heights = [-8, -23, -29, -37, -47, -55, -47, -46, -49, -57];
0.04
0
Original
Distorted
THD

10

23 / 10 
10

29 / 10 
10

37 / 10
10

8 / 10
  
10

57 / 10
-20
-40 f o
= 200 Hz
THD = 4.13%
-60
-80
-100
0 500 1000 1500 2000
Hz
2500 3000 3500 4000

Given the transfer characteristic for a class AB amplifier below, compute the THD for a 3 volt input sinusoid.
7v
V out
-3v
-1.75v
1.75v
3v
V in
-7v

-40
-60
3
Original
Distorted 2
1
0
-1
-2
-20
0
-3
0 0.005
0.01
0.015
0.02
seconds
0.025
0.03
0.035
0.04
Harmonic Peak Heights = [-7, -21, -46, -37, -44, -49, -45, -72, -49, -55];
Original
Distorted
THD

10

21 / 10 
10

46 / 10 
10

37 / 10
10

7 / 10
  
10

55 / 10 f o
= 200 Hz
THD = 4.14%
-80
-100
0 500 1000 1500 2000
Hz
2500 3000 3500 4000