UNIVERSITY OF CALIFORNIA AT BERKELEYDecimation and InterpolationDecimation• Anti-aliasing (decimation) filtering before downsampling• Filter has cutoff frequency of π/MInterpolation• Anti-imaging (interpolation) filtering after upsampling• Filter has cutoff frequency of π/LMh[n]L g[n]UNIVERSITY OF CALIFORNIA AT BERKELEYRational Rate ChangersChange the Sampling Rate by a Factor of L/M• Rational decimation system• General structure• f[n] is a lowpass filter with cutoff frequency• Film to NTSC format requires a 30/24 = 5/4 rate change• Speech compression from 48 KHz to 8 KHz requires a ratechange of 1/6, so there is no upsampler• What about CD to DAT conversion? 480/441?L Mf[n]fsL fsL fsL fs / MSamplingRateminπL---πM-----,UNIVERSITY OF CALIFORNIA AT BERKELEYUpsamplingcontinuous timediscrete timeupsampling by 3written as3 or 1:3Upsampling by L• Takes one sample andinserts L-1 zeroes after it• Increase sampling rate by L• Time domain• Frequency domain• Frequency axis expandedby a factor of Lyn[]xnL---ifnL---I∈0otherwise=Y ω() XLω()=UNIVERSITY OF CALIFORNIA AT BERKELEYDownsamplingcontinuous timediscrete timedownsampling by 3written as3 or 3:1Downsampling by M• Takes in M samples andoutputs the first sample• Reduces sampling rate by M• Time domain• Frequency domain• Frequency axis compressedby a factor of M• M-1 aliasing vectorsyn[] xMn[]=Yω() Xω 2πk–M-------------------k0=M1–∑=UNIVERSITY OF CALIFORNIA AT BERKELEYResamplingChanging the Sampling Rate• Conversion between audio formats• Speech compression• Video format conversionDigitalAudio TapeCompactDisc44.1 KHz 48 KHz48.044.1----------16---Speechon DAT48 KHzSpeech forTelephone8 KHzFilm Television24 frames/sec 30 frames/sec3024------UNIVERSITY OF CALIFORNIA AT BERKELEYAnalog-to-Digital Conversion1T---ynT()xt()yt()h(t) Q[.]yˆnT()Anti-AliasingFilterSampler Quantizerny(t)y(nT)xt()ttny(nT)^UNIVERSITY OF CALIFORNIA AT BERKELEYDisadvantages of Digital SystemsProgrammability• Speed is too slow for some applications• High average power and peak power consumptionRISC (2 Watts) vs. DSP (50 mW)• Aliasing from undersampling• Clipping from quantizationDATAMEMORYPROGMEMORYHARVARD ARCHITECTUREvQ[v]vUNIVERSITY OF CALIFORNIA AT BERKELEYAdvantages of Digital SystemsProgrammability• Modifiable in the field• Implement multiple standards• Better user interfaces• Tolerance for changes in specifications• Get better use of hardware for low-speed operations• Debugging• User programmabilityUNIVERSITY OF CALIFORNIA AT BERKELEYAdvantages of Digital SystemsPerfect reconstruction of asignal is possible even aftersevere distortionBetter trade-off betweenbandwidth and noiseimmunitybandwidthperformanceanalogdigitalIncrease signal-to-noiseratio simply by addingmore bitsSNR = -7.2 + 6 dB/bitUNIVERSITY OF CALIFORNIA AT BERKELEYAliasingfs-fs• Aliasing distortion• Quantization noise• Bandwidth limitations• Cost of A/D & D/A conversion• A 1 Hz Sine wave sampledat 1.8 Hz• A 0.8 Hz sine wave sampledat 1.8
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