Lecture 6: Modifying Sampled SignalsQuestionsGoals for TodayCreating a Signal in MATLABTime-domain view of signal recoveryReading a Signal into MATLABModifying SoundsSimple Signal OperationsSimple Signal OperationsTime ScalingFiltering: Changing Frequency ContentSpeech ExampleExample: Castanets & GuitarSquare Wave ExampleDifferent Varieties of Filters More on FilteringFor the ambitious….More on Time ScalingResampling vs. Time ScalingFiltering in MATLABLecture 6: Modifying Sampled SignalsThe Digital World of MultimediaProf. Mari OstendorfEE299 Lecture 618 Jan 2008Questions Do you want solutions posted for Lab 1? yes Extra lab hours – what times are good?(We have an undergrad assistant starting next week.) Weds after class?? (tbd) My office hours: Friday change to 9-11 Extra office hours – Thurs after 3:30 (tbd)EE299 Lecture 618 Jan 2008Goals for Today Working with digital signals in MATLAB Simple methods for modifying sounds Filtering Signal generationEE299 Lecture 618 Jan 2008Creating a Signal in MATLAB Desired signal: x(t)=cos(2πF0t), F0=100Hz Pick a sampling rate, say Fs=250Hz (Fs=250;Ts=1/Fs;) Corresponds to Ts=1/250 sec To make a 3 sec signal:  Create a time vector: t3 = [0:Ts:3]; Use that in the cosine function to get a value for x(n) at each time nTs: In math: x(n)= cos(2πF0nTs) In MATLAB: x= cos(2*pi*100*t3); To play the signal, use:sound(x,Fs) NOT sound(x). Why?stem(x) plots samplesplot(x) connects the dotsIf Fs is big enough, why doesn’t plot(x) look like cosine above?Need Fs to get back to the analog world.If not specified, Fs=8192Hz is used.See next slide….EE299 Lecture 618 Jan 2008Time-domain view of signal recovery Straight line interpolation is like putting a triangle at each sample and adding them up Using a smoother function works better (the picture is an approximation) In the frequency domain, it is an ideal low pass filterEE299 Lecture 618 Jan 2008Reading a Signal into MATLAB Demo: What’s wrong with this? haha = wavread(‘haha2’);  sound(haha); Right approach(es)… [haha, Fs, bits] = wavread(‘haha2’); or [haha, Fs] = wavread(‘haha2’); sound(haha,Fs);signal sampling rate quantizationEE299 Lecture 618 Jan 2008Modifying Sounds Amplitude operations Multiplying signals (e.g. for fading) Adding two signals (mixing) Time operations Time reverse Time delay Time-scale modification (speed up, slow down) Changing frequency content: filteringEE299 Lecture 618 Jan 2008Simple Signal Operations Amplitude scaling:y(n) = 3*x(n)Changes loudness, especially useful if you filter out frequency content and end up with a lower energy signal (watch out for clipping) Mixing signals: addition (optionally with scaling)y(n) = x1 (n) + 0.2*x2 (n) Changing the envelope, as in fading: multiplicationy(n) = x1 (n)*x2 (n) (use y=x1.*x2 in MATLAB)EE299 Lecture 618 Jan 2008Simple Signal Operations Time reverse (flipud(x) or fliplr(x) in MATLAB) Examples: music speech Time Delay: y(n) = x(n-T0) in MATLAB: delayed_x = [zeros(T0,1);x]; Echo: (Multiple time delays added)y(n) = x(n) + 0.5*x(n-T0 ) + 0.25*x(n-2T0 ) MATLAB example: relax>> pause=zeros(1500,1);>> echo=[relax;pause;relax*.4;pause;relax*.16;pause;relax*.05];>> sound(echo)EE299 Lecture 618 Jan 2008Time ScalingDrop samples:• speed up in time• higher pitch • higher frequency content(watch out for aliasing!)Insert samples: • slow down in time• lower pitch• lower frequency content(no aliasing worries)2X: Drop in MATLAB:for i=1:len/2cos2(i)=cos1(2*i-1);end0.5X: Insert in MATLAB:for i=1:len-1cos3(2*i-1)=cos1(i);cos3(2*i)=0.5*(cos1(i)+cos(i+1));endX(2n)X(n/2)EE299 Lecture 618 Jan 2008Filtering: Changing Frequency Content Low pass filter: Keeps low frequencies, suppresses high frequencies Smoothing effect High pass filter: Keeps high frequencies, suppresses low frequencies Emphasizes abrupt changes Band pass filter: Keeps some mid range of frequencies, suppresses both high and low frequenciesEE299 Lecture 618 Jan 2008Speech ExampleLow Pass FilteredHigh Pass FilteredOriginal SignalEE299 Lecture 618 Jan 2008Example: Castanets & GuitarEE299 Lecture 618 Jan 2008Square Wave ExampleOriginal signallow pass filteredhigh pass filteredEE299 Lecture 618 Jan 2008Different Varieties of Filters Different LPFsAll with cut-off at 100HzEE299 Lecture 618 Jan 2008More on Filtering Can also be used to “shape” frequency content Any function of the formy(n)=a1 y(n-1)+a2 y(n-2)+…+ap y(n-p)+b0 x(n)+b1 x(n-1)+…+bm (n-m)can be used to shape/change frequency content (input is x(n), output is y(n)) Leads us to the idea of generating signalsEE299 Lecture 618 Jan 2008For the ambitious….EE299 Lecture 618 Jan 2008More on Time Scaling What can you to avoid aliasing when increasing pitch? Low pass filter (LPF) the sound before you time scale Example:  Fs=16kHz, signal uses full 8kHz band,  want to time scale by a factor of 2 LPF at 4kHz cutoff (0.5 cutoff in digital filter design) Can you change the pitch without changing the duration? Yes, but it requires tricky cutting/pasting of pieces of the signal (beyond the scope of this class)EE299 Lecture 618 Jan 2008Resampling vs. Time Scaling Playing a signal with a different Fs has the same effect as time scaling, BUT It’s not a solution if you want to do multiple time scale changes within one sound file Time scaling: add/drop samples, keep Fs the same Resampling: changing Fs but NOT changing the duration/pitch of the signal Needed for combining signals of different sampling rates, or for taking up less disk space Resampling: add/drop samples, change Fs When decreasing Fs, watch out for aliasing! Handy function for resampling….EE299 Lecture 618 Jan 2008Filtering in MATLAB You can implement a filter in MATLAB by: writing a for loop to implement the system function (as in Lab2 “filtersounds.m”) Calling the built-in “filter” function, as in:y=filter(b,a,x) where b=[b0 b1 …bm ], a=[1 a1 a2 …ap ]  You can design a filter… By hand, as in: a=1; b=[1 1 1]/3; Using filter design commands, such as:[b a] = butter(n,Wn,’ftype’) OR b= fir1(n,Wn,'ftype');a=1;low, high, or stopcutoff(s):Ex: Wn=0.5=4k/8kFor 4k analog cutoff of signal with 8k range (Fs=16k)order (size of