Analog Vs. Digital Analog And Digital Signals - About

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Analog vs. DigitalAnalog and Digital SignalsIIIntroduction to DSP Systemsanalog signal continuous-time continuous amplitudedigital signal discrete-time discrete amplitudecontinuous amplitudediscrete amplitudex(t)Dr. Deepa Kundurx(t)22continuous-time-4University of Toronto-3-2-11234t10.5-4-3-2-10.5-2Dr. Deepa Kundur (University of Toronto)Introduction to DSP Systems1 / 30Dr. Deepa Kundur (University of Toronto)Analog vs. DigitalAnalog signals are fundamentally significant because we mustinterface with the real world which is analog by nature.Introduction to DSP Systems4t0123nIntroduction to DSP Systems1-3-2-101213n2 / 30analog system analog signal input analog signal outputIDigital signals are important because they facilitate the use ofdigital signal processing (DSP) systems, which have practicaland performance advantages for several applications.Dr. Deepa Kundur (University of Toronto)3Analog and Digital SystemsII2.5Analog vs. DigitalAnalog and Digital tages: easy to interface to real world, do not need A/D orD/A converters, speed not dependent on clock ratedigital system digital signal input digital signal outputI3 / 30advantages: re-configurability using software, greater controlover accuracy/resolution, predictable and reproducible behaviorDr. Deepa Kundur (University of Toronto)Introduction to DSP Systems4 / 30

Analog vs. DigitalAnalog vs. DigitalAnalog-to-Digital ConversionAnalog-to-Digital ConversionA/D converterxa(t)A/D gnalSampling:I conversion from cts-time to dst-time by taking “samples” atdiscrete time instantsI E.g., uniform sampling: x(n) xa (nT ) where T is the samplingperiod and n ZDr. Deepa Kundur (University of Toronto)Introduction to DSP Systems5 / 30Dr. Deepa Kundur (University of Toronto)Analog vs. DigitalAnalog-to-Digital ConversionA/D converterAnalogsignalA/D italsignalQuantization:I conversion from dst-time cts-valued signal to a dst-timedst-valued signalI quantization error: eq (n) xq (n) x(n) for all n Zx[n]Dr. Deepa Kundur (University of Toronto)6 / 30Analog vs. DigitalAnalog-to-Digital Conversionxa(t)Introduction to DSP Systems23Introduction to DSP Systemsn7 / 30Dr. Deepa Kundur (University of Toronto)Introduction to DSP Systems8 / 30

Analog vs. DigitalAnalog vs. DigitalAnalog-to-Digital ConversionAnalog-to-Digital ConversionA/D converterxa(t)A/D talsignalAnalogsignalQuantization:xq[n]-3Dr. Deepa Kundur (University of iscrete-timesignal9 / 30Dr. Deepa Kundur (University of Toronto)Analog vs. gnalCoderQuantizedsignalDigitalsignalIntroduction to DSP Systems10 / 30Sampling TheoremA/D converterx(n)01011.Analog vs. DigitalAnalog-to-Digital Conversionxa(t)xq(n)Coding:I representation of each dst-value xq (n) by ab-bit binary sequenceI e.g., if for any n, xq (n) {0, 1, . . . , 6, 7}, then the coder mayuse the following mapping to code the quantized amplitude:nIntroduction to DSP Systemsx(n)xq(n)If the highest frequency contained in an analog signal xa (t) isFmax B and the signal is sampled at a rate01011.CoderQuantizedsignalFs 2Fmax 2BDigitalsignalthen xa (t) can be exactly recovered from its sample values using theinterpolation functionExample coder:0123Dr. Deepa Kundur (University of Toronto)0000010100114567Introduction to DSP Systemsg (t) 100101110111sin(2πBt)2πBtNote: FN 2B 2Fmax is called the Nyquist rate.11 / 30Dr. Deepa Kundur (University of Toronto)Introduction to DSP Systems12 / 30

Analog vs. DigitalAnalog vs. DigitalSampling TheoremDigital-to-Analog ConversionSampling Period T 11 FsSampling Frequencyx[n]original/bandlimitedinterpolated signal1Therefore, given the interpolation relation, xa (t) can be written asxa (t) -3 X-2-1012n3xa (nT )g (t nT )n xa (t) I Xx(n) g (t nT )n Iwhere xa (nT ) x(n); called bandlimited interpolation.Dr. Deepa Kundur (University of Toronto)Introduction to DSP Systems13 / 30Common interpolation approaches: bandlimited interpolation,zero-order hold, linear interpolation, higher-order interpolationtechniques, e.g., using splinesIn practice, “cheap” interpolation along with a smoothing filteris employed.Dr. Deepa Kundur (University of Toronto)Analog vs. Digitalzero-orderholdIIDigital-to-Analog Conversionoriginal/bandlimitedinterpolated signal1-3T -2T -T0T2Tt3TCommon interpolation approaches: bandlimited interpolation,zero-order hold, linear interpolation, higher-order interpolationtechniques, e.g., using splinesIn practice, “cheap” interpolation along with a smoothing filteris employed.Dr. Deepa Kundur (University of Toronto)14 / 30Analog vs. DigitalDigital-to-Analog Conversionoriginal/bandlimitedinterpolated signalIntroduction to DSP SystemsIntroduction to DSP Systems15 / 30-3T -2T -TIIlinearinterpolation10T2Tt3TCommon interpolation approaches: bandlimited interpolation,zero-order hold, linear interpolation, higher-order interpolationtechniques, e.g., using splinesIn practice, “cheap” interpolation along with a smoothing filteris employed.Dr. Deepa Kundur (University of Toronto)Introduction to DSP Systems16 / 30

DSP SystemsDSP SystemsA DSP SystemA DSP SystemDSP SystemDSP SystemAntialiasingFilterAnalogsignalISampleand HoldAnalogsignalSampled lterReconstructionD/ADSPA/DFilterCts-time dst-amp“staricase” signalAnalogsignalIn practice, a DSP system does not use idealized A/D or D/Amodels.Dr. Deepa Kundur (University of Toronto)Introduction to DSP Systems17 / 30AnalogsignalSampleand HoldAnalogsignalSampled structionFilterCts-time dst-amp“staricase” signalAnalogsignalAnti-aliasing Filter:I ensures that analog input signal does not contain frequencycomponents higher than half of the sampling frequency (to obeythe sampling theorem)I this process is irreversibleDr. Deepa Kundur (University of Toronto)DSP SystemsIntroduction to DSP Systems18 / 30DSP SystemsA DSP SystemA DSP SystemInput SignalDSP -2Anti-aliased Signal-3-2-112Sampled structionFilterCts-time dst-amp“staricase” signalAnalogsignalSample and Hold:I holds a sampled analog value for a short time while the A/Dconverts and interprets the value as a digital2-4AnalogsignalSampleand Hold34t-2Dr. Deepa Kundur (University of Toronto)Introduction to DSP Systems19 / 30Dr. Deepa Kundur (University of Toronto)Introduction to DSP Systems20 / 30

DSP SystemsDSP SystemsA DSP SystemA DSP SystemAnti-aliased SignalDSP -2Sampled Data Signal2-4-3-2-1234AnalogsignalSampled structionFilterCts-time dst-amp“staricase” signalAnalogsignalA/D:I converts a sampled data signal value into a digital number, inpart, through quantization of the amplitudeanti-aliased signal1Sampleand Holdt-2Dr. Deepa Kundur (University of Toronto)Introduction to DSP SystemsDr. Deepa Kundur (University of Toronto)21 / 30DSP SystemsIntroduction to DSP Systems22 / 30DSP SystemsA DSP SystemA DSP SystemSampled Data Signal2-4-3-2-1DSP Systemanti-aliased signal1234AntialiasingFiltertAnalogsignal-2Digital Signal2-4-3-2-1234Introduction to DSP terCts-time dst-amp“staricase” signalAnalogsignalt-2Dr. Deepa Kundur (University of Toronto)Sampled datasignalD/ADSPA/DD/A:I converts a digital signal into a “staircase”-like signalsampled data signal1AnalogsignalSampleand Holdx(t)223 / 301Dr. Deepa Kundur (University of Toronto)Introduction to DSP Systems24 / 30

DSP SystemsDSP SystemsA DSP SystemA DSP SystemDigital Signal2-4-3-2-1DSP Systemsampled data se Signal2-3-2-11-2Dr. Deepa Kundur (University of Toronto)Sampled datasignal234Introduction to DSP SystemsDr. Deepa Kundur (University of Toronto)25 / 30Introduction to DSP Systems26 / 30Q: What are initial considerations when designing a DSP system thatmust run in real-time?2I-2-11234tI-2sampled data signalReconstructed Signal2-2-1Ianti-aliased signal1234It-2Dr. Deepa Kundur (University of Toronto)AnalogsignalReal-time DSP Considerationsdigital signal-3Cts-time dst-amp“staricase” signalDSP SystemsStaircase Signal-4DigitalsignalFiltersampled data signalA DSP System-3DigitalsignalReconstructiontDSP Systems-4D/ADSPA/DReconstruction Filter:I converts a “staircase”-like signal into an analog signal throughlowpass filtering similar to the type used for anti-aliasingdigital signal-4AnalogsignalSampleand HoldIntroduction to DSP Systemsx(t)227 / 301Algorithm: related to computational operations and accuracyrequired by the applicationSample rate: the rate at which input samples are received forprocessingSpeed: to meet an application throughput requirement with agiven sample rate, it must be possible to operate the DSP at aparticular speedNumeric representation: format and number of bits used fordata representation; depends on required computationalprecision and dynamic range required for applicationDr. Deepa Kundur (University of Toronto)Introduction to DSP Systems28 / 30

DSP SystemsDSP SystemsReal-time DSP ConsiderationsProgrammable DSPsQ: Is a DSP technology suitable for a real-time application?IIClock rate: rate at which a DSP performs its most basic unit ofwork; to meet the timing requirement with a given sampling rate, itmust be possible to operate the DSP at a particular clock rateIApplication-specific: designed to perform one function moreaccurately, faster or more cost-effectivelyIIThroughput: rate of multiply and accumulates (MACs) performed;measured in number of MACs per secondIGeneral purpose: microprocessor whose architecture is optimizedto process sampled data at high rates via pipelining andparallelismIArithmetic and addressing capability: requirements related to thealgorithm complexity, precision and data accessIPrecision: associated with format (fixed vs. floating), number of bitsIused for data representation, and required dynamic rangeIIexamples: FFT chips, digital filterscan be programmable within confines of a function; e.g.,coefficients of a digital filterprogrammable and more cost-effective for general computingshort system design cycle time Size, cost and power consumption: technology-dependentDr. Deepa Kundur (University of Toronto)Introduction to DSP Systems29 / 30Dr. Deepa Kundur (University of Toronto)Introduction to DSP Systems30 / 30

A DSP System A/D DSP D/A Analog signal Analog signal Sampled data signal Analog signal Cts-time dst-amp staricase signal Digital signal Digital signal DSP System Antialiasing Filter Sample and Hold Reconstruction Filter A/D: Iconverts a sampled data signal value into a digital number, in part, through quantization of the amplitude

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