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Basics on Digital Signal Processing Analog to Digital Conversion Oversampling or Σ-Δ converters Vassilis Anastassopoulos Electronics Laboratory, Physics Department, University of Patras Outline of the Lecture Analog to Digital Conversion - Oversampling or Σ-Δ converters Sampling Theorem - Quantization of Signals PCM (Successive Approximation, Flash ADs) DPCM - DELTA - Σ-Δ Modulation Principles - Characteristics 1st and 2nd order Noise shaping - performance Decimation and A/D converters Filtering of Σ-Δ sequences 2/39 Analog & digital signals Analog Digital Continuous function V Discrete function Vk of of continuous variable t Sampled discrete sampling (time, space etc) : V(t). Signal variable tk, with k = integer: Vk = V(tk). 0.3 0.3 0.2 0.2 0.1 0.1 0 0 Voltage [V] Voltage Voltage [V] Voltage Voltage [V] Voltage -0.1 -0.1 ts ts -0.2 -0.2 0 2 4 6 8 10 0 2 4 6 8 10 time [ms] sampling time, tk [ms] Uniform (periodic) sampling. Sampling frequency fS = 1/ tS 3/39 AD Conversion - Details 4/39 Sampling 5/39 Rotating Disk How fast do we have to instantly stare at the disk if it rotates with frequency 0.5 Hz? 6/39 1 The sampling theorem A signal s(t) with maximum frequency fMAX can be Theo* recovered if sampled at frequency fS > 2 fMAX . * Multiple proposers: Whittaker(s), Nyquist, Shannon, Kotelnikov. Naming gets confusing ! Nyquist frequency (rate) fN = 2 fMAX or fMAX or fS,MIN or fS,MIN/2 Example s(t) 3cos(50πt) 10sin(300πt) cos(100πt) Condition on fS? F1 F2 F3 f > 300 Hz F1=25 Hz, F2 = 150 Hz, F3 = 50 Hz S fMAX 7/39 Sampling and Spectrum 8/39 1 Sampling low-pass signals Continuous spectrum (a) (a) Band-limited signal: frequencies in [-B, B] (fMAX = B). -B 0 B f (b) Discrete spectrum No aliasing (b) Time sampling frequency repetition. fS > 2 B no aliasing. -B 0 B fS/2 f Discrete spectrum (c) Aliasing & corruption (c) fS 2 B aliasing ! Aliasing: signal ambiguity 0 fS/2 f in frequency domain 9/39 Quantization and Coding N Quantization Levels q Quantization Noise 10/39/ 2 SNR of ideal ADC RMSinput Assumptions SNR 20log (1) ideal 10 RMS(eq) Ideal ADC: only quantisation error eq (p(e) constant, no stuck bits…) Also called SQNR (signal-to-quantisation-noise ratio) eq uncorrelated with signal (noise spectrum???) ADC performance constant in time. T 2 1 V V Input(t) = ½ V sin( t). RMSinput FSR sinωt dt FSR FSR T 2 2 2 0 p(e) quantisation error probability density q/2 2 q VFSR 1 RMS(eq) eq peqdeq q 12 N -q/2 2 12 q q (sampling frequency f = 2 f ) eq S MAX 2 2 Error value 11/39/ 2 SNR of ideal ADC - 2 Substituting in (1) : SNRideal 6.02N1.76 [dB] (2) One additional bit SNR increased by 6 dB Real SNR lower because: - Real signals have noise. - Forcing input to full scale unwise. - Real ADCs have additional noise (aperture jitter, non-linearities etc). Actually (2) needs correction factor depending on ratio between sampling freq & Nyquist freq. Processing gain due to oversampling. 12/39 Coding – Flash AD 13/39 Coding - Conventional 14/39 Digital Filters •They are characterized by their Impulse Response h(n), their Transfer Function H(z) and their Frequency Response H(ω). •They can have memory, high accuracy and no drift with time and temperature. •They can possess linear phase. •They can be implemented by digital computers. 15/39 Digital Filters - Categories N M IIR y(n) a(k)x(n k) bk y(n k) k0 k1 N k ak z x(n) x(n-1) x(n-2) x(n-Ν+1) x(n-Ν) Ts Ts Ts k0 H(z) M 1 b z k k a a a a a k1 0 1 2 N-1 N y(n) + bΜ bΜ-1 b2 b1 T T T y(n-Μ) s y(n-Μ+1) y(n-2) s y(n-1) s 16/39 Digital Filters - Categories N N y(n) h(k)x(n k) a(k)x(n k) FIR k0 k0 N N k k H(z) ak z h(k)z k0 k0 x(n) x(n-1) x(n-2) x(n-Ν+2) x(n-Ν+1) Ts Ts Ts aΝ-2 ή a ή a0 ή a1 ή a2 ή Ν-1 h(0) h(1) h(2) h(Ν-2) h(Ν-1) y(n) + •Stable •Linear phase 17/39 Digital Filters - Examples a a z 1 a z 2 0 1 2 1.5 1 H(z) 1 2 1 b1z b2 z 0 1 a0=0.498, a1=0.927, -1 a2=0.498, b1=-0.674 b2=-0.363. 0.5 IIR Phase Magnitude IIR Magnitude -2 0 -3 0 0.1 0.2 0.3 0.4 0 0.1 0.2 0.3 0.4 Frequency Frequency 11 1.5 4 H(z) h(k)z k 2 k0 1 0 h(1)=h(10)=-0.04506 0.5 FIR Phase h(2)=h(9)=0.06916 FIR Magnitude -2 h(3)=h(8)=-0.0553 0 -4 0 0.1 0.2 0.3 0.4 0 0.1 0.2 0.3 0.4 h(4)=h(7)=-.06342 Frequency Frequency h(5)=h(4)=0.5789 18/39 FIR filters N N y(n) h(k)x(n k) a(k)x(n k) k0 k0 N N k k H(z) ak z h(k)z k0 k0 x(n) x(n-1) x(n-2) x(n-Ν+2) x(n-Ν+1) Ts Ts Ts aΝ-2 ή a ή a0 ή a1 ή a2 ή Ν-1 h(0) h(1) h(2) h(Ν-2) h(Ν-1) Design Methods y(n) •Stable + • Optimal filters • Windows method •Linear phase • Sampling frequency 19/39 Linear phase - Same time delay 180o 3*180o 5*180o If you shift in time one signal, then you have to shift the other signals the same amount of time in order the final wave remains unchanged. For faster signals the same time interval means larger phase difference. Proportional to the frequency of the signals. -αω 20/39 Oversampling and Noise Shaping Converters 21/39 Quantization Noise Shaping 22/39 Differential Pulse Code Modulation - DPCM 23/39 Delta Modulation – One bit DPCM The coder is an integrator as the one in the feedback loop. 24/39 Delta Sequences 25/39 ΔΣ - Encoder 26/39 ΔΣ – Encoder Behavior - Signal Transfer Function 27/39 ΔΣ – Encoder Behavior Quantization Noise Signal Transfer Function (X=0) 28/39 Quantization Noise Power Density in the Signal Band 29/39 A Second Order ΔΣ Modulator 30/39 Third Order Noise Shaper 31/39 Digital Decimation 32/39 Digital Decimation – ΣΔ to PCM Conversion 33/39 Digital Decimation – ΣΔ to PCM Conversion 34/39 Multi-Rate FIR filters 35/39 Decimation Low rate sequences are finally stored on CDs 36/39 Interpolation Interpolation is used to convert the Low rate sequences stored on the CDs to analog signals Oversampling is used 37/39 Oversampling in Audio Systems 38/39 39/39 .
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