1. Research Methods in Acoustics Lecture 8: Digital systems and filters
Jonas Braasch

2. Orthogonality
Orthogonal (greek: orthos=straight/right, gonia=angle) is often used as synonym for perpendicular or right-angled.
Two at 90° angle crossing streets are orthogonal to each other
The dot-product of two vectors is zero if both vectors are orthogonal to each other.

3. Orthogonality
Vector in 2- or 3-dimensional Euclidean space

4. Orthogonality Vector in 2- or 3-dimensional Euclidean space
Orthogonal case:

5. Orthogonality Vector in 2- or 3-dimensional Euclidean space angle:
Length of vector
angle=1 parallel
angle=0 orthogonal
angle=-1 anti-parallel
A unit vector has the length of one.

6. Orthogonality
angle between two vectors
angle between n·m

7. Orthogonality Vector in 2- or 3-dimensional Euclidean space a) a) b)
90°
90° b) c) c) d)
90° d)
45°

8. Digital Signals (Sampling)
samples
The function y(t) is a continuous (analog) signal that we would like to transform into a digital signal.
We can do this by measuring the value of the continuous signal at constant time intervals n·T (with n=0,1,2,3,…):
Fs=48 kHz
f=1000 Hz

9. Sampling Rate
The sampling frequency (sampling rate fs or Fs) is defined as the number of samples that are recorded during a given time interval. The unit is typically [fs]=1/s or Hz.
The sampling frequency is the inverse of the sampling interval T after which we recorded the next sample.
Fs=48 kHz, f=1000 Hz, T=0.1 ms T
time [s]

10. Example: Fs=48 kHz, f=1000 Hz
time [s]
samples

11. Fs=48 kHz, signal 30 kHz
Fs=48 kHz, signal 20 kHz

12. Quantization
In a digital signal, each sample is quantized to a new integer value.
In computer systems, we transmit information using and on/off code instead of having decimal digits from 0−9 (Base 10). Internally, the computer only knows two digits 0 and 1 (Base 2). It operates on a binary code.
One digit, which can be either 0 or 1 is called a bit. A group of 8 bits is referred to as one byte.
We can easily transform a binary number into a number to the base 10:
With x10 the number to the base 10 and x2,n the nth digit of the base-2 number.
Example: x2=1011

13. Quantization
The sampling frequency (sampling rate fs or Fs) is defined as the number of samples that are recorded during a given time interval. The unit is typically [Fs]=1/s or Hz.
The sampling frequency is the inverse of the sampling interval T after which we recorded the next sample.
20*log10(256/2) =20*log10(2^8/2) =
20*log10(4096/2) =20*log10(2^12/2) =
20*log10(65536/2) =20*log10(2^16/2) =
20*log10( /2) =20*log10(2^24/2) =
Divide by 2, because one bit is
lost for the sign (plus or minus)

14. Conversion: Binary to Decimal
function [y]=convbin(x)
numofdigits=floor(log2(x));
for n=numofdigits+1:-1:1
y(n)=mod(x,2);
x=floor(x./2);
end % of for

15. Binary Demo Demo Results: zdB =−10.6060 dB ydB = 31.4045 dB
x=randn(1000,1);
x=abs(x);
x=(128.*x./max(abs(x)));
x0=round(x);
for n=1:length(x0)
y2=convbin(x0(n));
y(n)=convBase10(y2);
end % of for
z=y-x';
% level of signal
yr=sqrt(mean(y.^2));
ydb=20*log10(yr)
% quantization error dB
zr=sqrt(mean(z.^2));
zdb=20*log10(zr)
Demo Results:
zdB =− dB
ydB = dB
ydB−zdB = dB
Compare to the theoretical result of
20*log10(2^8/2)= dB

16. Theorems
Nyquist–Shannon sampling theorem describes the conditions in which the continous signal can be perfectly reconstructed.
If the conditions of the Nyquist–Shannon sampling theorem are met, the sampled signal can be reconstructed using the Whittaker–Shannon interpolation formula.
Typically, the signal is bandwidth limited to fs/2 (Nyquist-Frequency). However, it is also possible to unambiguously code signals above this frequency, if the continuous signal is limited to a corresponding low-frequency cut-off frequency. This is often useful in radio broadcast.

17. Aliasing
The effect that two different continuous signals to become indistinguishable when sampled is called aliasing.
Aliasing typically occurs when a signals frequency is more the than twice the sampling frequency (see red curve). It then becomes an alias of a lower frequency (blue curve).
An anti-aliasing filter, which is a low-pass filter is used to avoid this phenomenon. All frequencies above fs/2 are eliminated before they are converted into a digital signal.

18. Fourier Transformation with e-function
Basically, we can also treat non-harmonic functions by setting T to ∞. We can also use the complex e-function instead of sine and cosine:
x=2pt, T→∞

19. Fourier Transformation of a sine signal
Fs=12000;
t=0:1./Fs:1; % create time signals
ffttabs=8182;
f=(1./ffttabs:1./ffttabs:1).*Fs;
x=sin(2*pi*1000*t);
y=fft(x,ffttabs);
plot(f,imag(y),'r:','LineWidth',2)
hold on
plot(f,real(y),'LineWidth',2)
hold off
ylabel('amplitude','FontSize',14);
xlabel('frequency [Hz]','FontSize',14);
set(gca, 'FontSize',14);

20. Samples vs. time and frequency
Fs=12000;
t=0:1./Fs:1; % create time signals
% choose 1./Fs as step size (equals one sample)
% A similar approach for the frequency space
ffttabs=8182; % number of frequency samples
f=(1./ffttabs:1./ffttabs:1).*Fs;
% We know that the whole frequency space ranges % from 0 to Fs

21. Fourier Transformation of sine signals
Imaginery
component
real
component
Real part, axis-symmetrical
Imaginery part, point-symmetrical

22. Fourier Transformation of a cosine signal
Fs=12000;
t=0:1./Fs:1; % create time signals
ffttabs=8182;
f=(1./ffttabs:1./ffttabs:1).*Fs;
x=sin(2*pi*1000*t);
x2=cos(2*pi*1000*t);
y2=fft(x2,ffttabs);
plot(f,real(y2),'LineWidth',2)
hold on
plot(f,imag(y2),'r:','LineWidth',2)
hold off
ylabel('amplitude','FontSize',14);
xlabel('frequency [Hz]','FontSize',14);
set(gca, 'FontSize',14);

23. Fourier Transformation of a cosine signal
real
component
Imaginery
component
Note, that now the real component
has the larger peak

24. Power spectrum
The amplitude of the Fourier signal can be estimated from the magnitude of the Fourier transformed signal H(w):
We can also determine the power signal
We used the following relationship

25. Power spectrum figure % Power spectrum % sine signal Pyy=y.*conj(y);
subplot(2,1,1);
plot(f,Pyy,'LineWidth',2)
ylabel('Power spectrum','FontSize',14);
xlabel('frequency [Hz]','FontSize',14);
set(gca, 'FontSize',14);
% cosine signal
Pyy2=y2.*conj(y2);
subplot(2,1,2);
plot(f,Pyy2,'r','LineWidth',2);

26. Power spectrum
sine
signal
cosine
signal

27. Phase
The phase of the signal is the argument of the spectrum:

28. Phase p1 = atan2(imag(y), real(y)); p2 = atan2(imag(y2), real(y2));
phase1=180/pi*unwrap(p1);
phase2=180/pi*unwrap(p2);
figure
plot(f,phase1,'LineWidth',2);
hold on
plot(f,phase2,'r:','LineWidth',2);
plot(f,phase2-phase1,'g','LineWidth',2);
hold off
ylabel('phase [deg]','FontSize',14);
xlabel('frequency [Hz]','FontSize',14);
set(gca, 'FontSize',14);

29. Phase Blue=sine red=cosine Green=phase difference
Phase difference 1000 Hz

30. Fourier transformation of an impulse
x=zeros(length(t),1);
x(1)=1;
figure
stem(x)
ylabel('amplitude','FontSize',14);
xlabel('time [samples]','FontSize',14);
axis([ ]);
set(gca, 'FontSize',14);

31. Impulse

32. Fourier transformation of an impulse
x=zeros(length(t),1);
x(1)=1;
x2=zeros(length(t),1);
x2(8)=1;
figure
stem(x)
ylabel('amplitude','FontSize',14);
xlabel('time [samples]','FontSize',14);
axis([ ]);
set(gca, 'FontSize',14);
stem(x2)

33. Impulse

34. Fourier Transformation
f=(1./ffttabs:1./ffttabs:1).*Fs;
y=fft(x,ffttabs);
y2=fft(x2,ffttabs);
figure
% Power spectrum
Pyy=y.*conj(y);
Pyy2=y2.*conj(y2);
plot(f,Pyy,'LineWidth',2)
hold on
plot(f,Pyy,'r--','LineWidth',2)
hold off
ylabel('Energy','FontSize',14);
xlabel('frequency [Hz]','FontSize',14);
axis([ ]);
set(gca, 'FontSize',14);

35. Magnitude Spectrum

36. Phase calculation p1 = atan2(imag(y), real(y));
phase1=180/pi*unwrap(p1);
p2 = atan2(imag(y2), real(y2));
phase2=180/pi*unwrap(p2);
figure
plot(f,phase1,'LineWidth',2)
hold on
plot(f,phase2,'r--','LineWidth',2)
hold off
ylabel('phase [deg]','FontSize',14);
xlabel('frequency [Hz]','FontSize',14);
axis([ ])
set(gca, 'FontSize',14);

37. Phase

38. Phase and Group delay
phase delay
group delay

39. Group delay figure domega=pi*Fs/length(y); R=real(y); I=imag(y);
dI=derive(I,domega);
dR=derive(R,domega);
grpdelay=-(R.*dI-I.*dR)./(abs(y).^2);
plot(f,grpdelay*Fs./2,'LineWidth',2)

40. Group delay (cont.) domega=pi*Fs/length(y2); R=real(y2); I=imag(y2);
dI=derive(I,domega);
dR=derive(R,domega);
grpdelay2=-(R.*dI-I.*dR)./(abs(y2).^2);
hold on
plot(f,grpdelay2*Fs./2,'r','LineWidth',2)
hold off
ylabel('group delay [in samples]','FontSize',14);
xlabel('frequency [Hz]','FontSize',14);
axis([ ])
set(gca, 'FontSize',14);

41. Group delay

42. Phase and Group delay 108° 72° Dy 36° 0° Dx 100 200 300 f [Hz]
Phase delay describes the overall delay
of the phase
Group delay describes the time delay
compare to adjacent frequencies
108°
tgr=dy/dx
72° Dy
36°
tph=Dy/Dx 0° Dx
100
200
300
f [Hz]
phase delay
group delay

43. What does linear phase mean?
T=1/f [ms] f=360° · t/T
108°
300 Hz
200 Hz
72°
100 Hz
36°
1ms

44. Linear phase 108° 72° 36° 0° 100 200 300 f [Hz]
Let us draw the values from the last slide:
constant delay<->linear phase
108°
72°
36° 0°
100
200
300
f [Hz]

45. Impulse Fs=12000; ffttabs=8182; x=FIR1(512,0.5,'low');
plot(x,'LineWidth',2);
ylabel('amplitude','FontSize',14);
xlabel('time [samples]','FontSize',14);
set(gca, 'FontSize',14);

46. FIR low-pass filter (300 Hz), n=512

47. Magnitude spectrum f=(0:1./(ffttabs-1):1).*Fs; y=fft(x,ffttabs);
figure
% Power spectrum
Pyy=y.*conj(y);
plot(f,Pyy,'LineWidth',2)
ylabel('Energy','FontSize',14);
xlabel('frequency [Hz]','FontSize',14);
axis([ ]);
set(gca, 'FontSize',14);

48. Magnitude spectrum

49. Phase p1 = atan2(imag(y), real(y)); phase1=180/pi*unwrap(p1); figure
plot(f,phase1,'LineWidth',2)
ylabel('phase [deg]','FontSize',14);
xlabel('frequency [Hz]','FontSize',14);
set(gca, 'FontSize',14);

50. Phase

51. Group delay figure domega=pi*Fs/length(y); R=real(y); I=imag(y);
dI=derive(I,domega);
dR=derive(R,domega);
grpdelay=-(R.*dI-I.*dR)./(abs(y).^2);
phasedelay=-unwrap(p1)./(2.*pi.*f);
plot(f,grpdelay*Fs./2,'LineWidth',2)
hold on
plot(f,1+phasedelay*Fs,'r','LineWidth',2)
hold off
ylabel('delay [in samples]','FontSize',14);
xlabel('frequency [Hz]','FontSize',14);
legend('group delay','phase delay');
%axis([ ])
%figure
%grpdelay(1,1)
set(gca, 'FontSize',14);

52. Phase and Group Delay

53. FIR low-pass filter (300 Hz), n=32

54. Magnitude spectrum

55. Phase

56. Group and phase delay

57. Linear plot

58. Log-frequency plot

59. Log-log plot
cut-off frequency
−3dB threshold

60. Homework 6, #1 % Pythagorean Tempering
function [freqPytha,freqEqual,devCent]=Interval2Freq(interval)
% Pythagorean Tempering
rFreq = 440; %Hz (A4) reference Frequency
fifth = 3/2;
% Determine intervals for first octave
for n=2:12
rFreq(n)=rFreq(n-1)*3/2;
if rFreq(n)>rFreq(1)*2
rFreq(n)=rFreq(n)/2;
end % of if
end % of for
rFreq=sort(rFreq);
octave=floor(interval./12);
chroma=mod(interval,12);
freqPytha=2.^octave.*rFreq(chroma+1);
% Equal Tempering
freqEqual=rFreq(1).*2.^(interval./12);
devCent=1200.*log2(freqPytha./freqEqual);

61. Results >> [freqPytha,freqEqual,devCent]=Interval2Freq([1:12])
freqEqual =
devCent =

62. Homework 6, #2
function [y,t]=OrganPipe(pipelength,reflectioncoef,signal)
c=343; % m/s
Fs=48000;
x=signal;
t=(0:length(signal)-1)./Fs;
PipeTabs=round(Fs*pipelength/c);
l=PipeTabs/Fs*c;
ForwardDelayLine=zeros(PipeTabs+1,1);
BackwardDelayLine=zeros(PipeTabs+1,1);
PipeTabs=PipeTabs+1;
for n=1:(length(x))
ForwardDelayLine(2:PipeTabs)=ForwardDelayLine(1:PipeTabs-1);
ForwardDelayLine(1)=-1*BackwardDelayLine(1)+x(n);
y(n)=ForwardDelayLine(1);
BackwardDelayLine(1:PipeTabs-1)=BackwardDelayLine(2:PipeTabs);
BackwardDelayLine(PipeTabs)=reflectioncoef*y(n);
end % of for

63. Result open pipe
[y]=OrganPipe(0.5,-0.7,[1; zeros(4800,1)]);

64. Result open pipe
[y,t]=OrganPipe(0.5,-0.7,[1; zeros(4800,1)]);

65. Result closed pipe
[y,t]=OrganPipe(0.5,0.7,[1; zeros(4800,1)]);

66. convolution x=randn(4800,1);
[y,t]=OrganPipe(0.5,0.7,[1; zeros(4800,1)]);
Z=conv(x,y);