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

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.

Orthogonality Vector in 2- or 3-dimensional Euclidean space

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

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.

Orthogonality angle between two vectors angle between n·m

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

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

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]

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

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

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

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) =42.1442 20*log10(4096/2) =20*log10(2^12/2) =66.2266 20*log10(65536/2) =20*log10(2^16/2) =90.3090 20*log10(16777216/2) =20*log10(2^24/2) =138.4738 Divide by 2, because one bit is lost for the sign (plus or minus)

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

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 =−10.6060 dB ydB = 31.4045 dB ydB−zdB = 42.0105 dB Compare to the theoretical result of 20*log10(2^8/2)=42.1442 dB

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.

Aliasing http://en.wikipedia.org/wiki/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.

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→∞

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);

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

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

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);

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

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

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);

Power spectrum sine signal cosine signal

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

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);

Phase Blue=sine red=cosine Green=phase difference Phase difference 90°@ 1000 Hz

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([0 12 0 1.2]); set(gca, 'FontSize',14);

Impulse

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([0 12 0 1.2]); set(gca, 'FontSize',14); stem(x2)

Impulse

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([0 12000 0 1.2]); set(gca, 'FontSize',14);

Magnitude Spectrum

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([0 12000 -2600 100]) set(gca, 'FontSize',14);

Phase

Phase and Group delay phase delay group delay

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)

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([0 12000 -1 8]) set(gca, 'FontSize',14);

Group delay

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

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

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]

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);

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

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([0 12000 0 1.2]); set(gca, 'FontSize',14);

Magnitude spectrum

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);

Phase

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([0 12000 -1 8]) %figure %grpdelay(1,1) set(gca, 'FontSize',14);

Phase and Group Delay

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

Magnitude spectrum

Phase

Group and phase delay

Linear plot

Log-frequency plot

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

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);

Results >> [freqPytha,freqEqual,devCent]=Interval2Freq([1:12]) 469.8633 495.0000 528.5962 556.8750 594.6707 626.4844 660.0000 704.7949 742.5000 792.8943 835.3125 880.0000 freqEqual = 466.1638 493.8833 523.2511 554.3653 587.3295 622.2540 659.2551 698.4565 739.9888 783.9909 830.6094 880.0000 devCent = 13.6850 3.9100 17.5950 7.8200 21.5050 11.7300 1.9550 15.6400 5.8650 19.5500 9.7750 0

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

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

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

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

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