Fundamentals of digital signal processing - unipd.it

1 1 Fundamentals of digitalsignalprocessing2 Sound modelingsoundsymbolsaimsanalysissynthesi sprocessingclassification: signal modelssource modelsabstract (t)t in s e c x(n) y(n) in s e c y(t) digital signalsanalogsamplingprocessingreconstru ctionsampling interval Tsampling frequency fs= 1/T4 digital signals: time representations8000 samples100 samplesline with dotsvertical -32768 .. -1 .. (1-Q)Q = quantization (n) (n) (n)n 5 Spectrum: analog vs. digital signalsamplingleadstoa replicationof the basebandspectrum6 Spectrum: analog vs. digital signalSampling leads to a replication of the analog signal spectrumReconstruction of the analog signal :low pass filtering the digital signal7 Discrete 104-40-20020|X(f)| in dB f in Hz Ma gnitude s pe c trum |X(f)| in dBDiscrete FourierTransform(example)FFT with 16 pointscosine (16 points)magnitude (16 points)normalization:0 dB for sinusoid 1magnitude (frequency points)step fs/Nmagnitude dB vs.

2 Hz0246810121416-101a )n Cos ine s igna l x(n) )k Ma gnitude s pe c trum |X(k)| )f in Hz Ma gnitude s pe c trum |X(f)|Nkfs/9 Inverse Discrete FourierTransform(DFT)ifX(k) = X*(N-k)thenIDFT givesN discrete-time realvaluesx(n)10 FrequencyresolutionZero padding: toincreasefrequencyresolution0246-1012x( n) 8 s a mple s02460246810|X(k)| 8-point FFT051015-1012n x(n) 8 s a mple s + z e ro-pa dding0510150246810k |X(k)| 16-point FFT11 Window functionstoreduce leakage:weightaudio samplesbya windowHammingwindowwH(n) = cos(2 n/N)BlackmanwindowwB(n) = (2 n/N) + (4 n/N)12 WindowReduction of the leakage effect by window functions: (a) the original signal , (b) the Blackman window function of length N =8, (c) product x(n)w(n) with 0 n N-i, (d) zero-padding applied to z(n) w(n) up to length N = 16 The corresponding spectra are shown on the right igna l x(n)x(n) n 051015200200040006000-100-500Wa te rfa ll Re pre s e nta tion of S hort-time FFTsf in Hz n Ma gnitude in dB 15 digital systems16 DefinitionsUnitimpulseImpulsereponseh(n) = output toa unitimpulseh(n)describesthe digitalsistemDiscrete convolution:y(n)=x(n)*h(n)17 Algorithmsand y(n) = x(n-2) y(n) = a x(n) (n) = a1 x(n) + a2 x(n)18 Simpledigitalsystemweightedsumover severalinput samples19 TransformsFrequencydomaindesciptionof the digitalsystemZ transformDiscrete time Fourier transformTransfer function H(z).

3 Z transform of h(n)Frequency response:Discrete time Fourier transformof h(n)20 Causaland stablesystemsCausality: a discrete-time system iscausalif the output signal y(n)= 0 for n <0 fora giveninput signalu(n) = 0 forn < means that the system cannot react to an input before the input is applied to the systemStability: a digital system is stable ifstability implies that transfer function H(z)and frequency response are related by21 IIR systems= system withinfinte secondorderIIR systemDifferenceequationTransfer function22 IIR systems= system withinfinte impulseresponseh(n)DifferenceequationZ transform of diff. function23 FIR systems= system withfinite impulseresponseh(n) secondorderFIR systemDifference equationZ transform of diff. function24 Firexamplecomputation of frequency responseimpulse responsemagnitude plotphase (a ) Impuls e Re s pons e h(n)n -1012-101Re (z )Im(z )(c ) P ole /Ze ro in kHz |H(f)| (b) Ma gnitude Re s pons e |H(f)| in kHz H(f)/ (d) P ha s e Re s pons e H(f)