Lecture 5: From Analog to Digital (cont.)AnnouncementsPlan for TodayAnalog vs. DigitalReminder of Key Sampling ResultIn the frequency domain…Hearing and Seeing AliasingHow to Avoid AliasingWhy are CDs sampled at 44.1kHz?QuantizationQuantization with a MapNext step: Use Binary NumbersBinary NumbersMore on Binary NumbersCombining Sampling & QuantizationLanguage is a signal too!Lecture 5: From Analog to Digital (cont.)The Digital World of MultimediaProf. Mari OstendorfReading: Orsak et al., Chapter 5EE299 Lecture 516 Jan 2008Announcements Assignments due this week:  Lab1 (Matlab intro) and HW1  Lab 2 is being updated to  ask for a CollectIt submission,  explain how to write out a sound file in Matlab fix a few typosNew version tonight.EE299 Lecture 516 Jan 2008Plan for Today Review main sampling result Frequency domain view of sampling Quantization Binary representationsEE299 Lecture 516 Jan 2008Analog vs. Digitalanalog signalx(t)digital signalx(n)tncontinuous in timediscrete in timeSAMPLINGcontinuous in amplitudediscrete in amplitudeQUANTIZATIONSound wave, heart beat, temperature fluctuation, image on film, …Audio file on a CD, signal on a digital computer, Dow Jones daily average, image from a digital cameraEE299 Lecture 516 Jan 2008Reminder of Key Sampling ResultTo represent an analog signal on a computer, and be able to recover it later, you need to sample at least twice as fast as the highest frequency: SampleA/DInterpolateD/AFs > 2Fctime signalx(t)sampling ratemax frequency in X(f)EE299 Lecture 516 Jan 2008In the frequency domain… Sampling effectively duplicates the frequency content of the signal (periodic in frequency) If the duplication isn’t spaced far enough apart, there’s overlap (or wraparound)SampleFs < 2FcFc-Fc……2FsFc-Fc-FsFs-2FsNote: Here we use complex exponentials instead of cosinescos(at)=0.5(ejat+e-jat)so we have positive & negative frequencyEE299 Lecture 516 Jan 2008Hearing and Seeing Aliasing Sound demo: higher frequencies sound lower, mixing with actual low frequencies can make it sound muffled See: sound_sampling.mEE299 Lecture 516 Jan 2008How to Avoid Aliasing Pick a desired sampling rate Fs Eliminate frequency content above Fc=Fs/2 with a low-pass filter Then sampleSampleA/DInterpolateD/AWFs > 2Fctime signalx(t)X(f)Anti-aliasingfilterFc < WZ(f)z(t)z(n)Note: We’ll learn about filters next lecture.EE299 Lecture 516 Jan 2008Why are CDs sampled at 44.1kHz? Sampling rate must be greater than 2*20kHz to get full hearing range of most listeners Since practical filters aren’t perfect, it’s a good idea to go a bit higher But why 44.1kHz exactly? Arguing over standards… inherited from a method for storing digital audio on analog video tapes. FcIdealfilterFcNon-idealfilterEE299 Lecture 516 Jan 2008Quantization Map continuous-valued amplitude of each sample into one of a finite set of numbers Levels can be… uniform -0.5, -0.4, -0.3, -0.2, -0.1, 0.0, 0.1, 0.2, 0.3, 0.4, 0.5 or arbitrary-0.5, -0.32, -0.2, -0.1, -0.04, 0, 0.04, 0.1, 0.2, 0.32, 0.5EE299 Lecture 516 Jan 2008Quantization with a Map Represent that number with an index (e.g. 0,1,…,255) Use that index with a table of values when you play the sound or display the image Leverages human perception: some regions (or frequencies) need fewer levels than othersQ(X)computer or communicationschannelV(I’)Xmap values VX’I I’EE299 Lecture 516 Jan 2008Next step: Use Binary Numbers 1/0 is easy to store, e.g. Up/down direction of magnetization Smooth (reflective) surface vs. pit for CD On/off in memory 1/0 distinction is good for reliability in communications, can detect signal and resend at repeater before noise gets too great On/off switch is easy (fast) for computing In general, the binary representation is simple and reliableEE299 Lecture 516 Jan 2008Binary Numbers Binary numbers d3 d2 d1 d0 in binary = d3x23+ d2x22+ d1x21 + d0x1 k bits let you represent 2knumbers 8 levels Î 3 bits (23=8) 8 bits Î 256 levels (28=256) Examples: 10 in decimal = 1010 in binary 10 in binary = 2 in decimal In binary: 1+1=10 Question: what is 56 in binary? 111000EE299 Lecture 516 Jan 2008More on Binary Numbers To represent both positive and negative numbers, you need an extra “sign” bit To represent non-integer numbers, use bits to the right of an assumed “point” for negative powers Decimal:  753.25 = 7x102+ 5x10 + 3x1 + 2x10-1+ 5x10-2 Binary (=5.75 in decimal) 101.11 = 1x22 + 0x2 + 1x1 + 1x2-1+ 1x2-2EE299 Lecture 516 Jan 2008Combining Sampling & Quantization Number of bits to store a signal Duration of audio signal (or size of image,…)X sampling rateX bits per sample (depends on # of quantizer levels) Example 1:  how many bits are required to store 1 minute of stereo CD music? (44.1k sample rate, 16 bit samples) (60 sec)x(44.1k samples/sec)x(16 bits/sample)x(2 channels) = 84.7 Mbits Example 2:  how precise is the quantization telephone speech? (64kbs with a sample rate of 8k) (64k bits/sec)/(8k samples/sec) = 8 bits/sampleEE299 Lecture 516 Jan 2008Language is a signal too! How do we represent language on a computer? Via characters: In English, ASCII mapping between 7-bit binary code and characters (letters, numbers, space, symbols)You & I, 1011001 1101111 1110101 0100000 0100110 01000000 1001001 0101100 In other languages (e.g. Chinese), a longer code is needed Via words: A word corresponds to an index in a map, useful for search engines, text analysis,