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) 3cos(50πt) 10sin(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

RMSinput Assumptions SNR  20log   (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). RMSinput    FSR sinωt  dt  FSR FSR T  2  2 2 0

p(e) quantisation error probability density

q/2 2 q VFSR 1 RMS(eq)  eq peqdeq   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  6.02N1.76 [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) k0 k1 N k ak z x(n) x(n-1) x(n-2) x(n-Ν+1) x(n-Ν)  Ts Ts Ts k0 H(z)  M 1 b z k  k a a a a a k1 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 k0 k0 N N k k H(z)  ak z  h(k)z k0 k0

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 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 k0 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 N N y(n)  h(k)x(n  k)  a(k)x(n  k) k k   H(z)  ak z  h(k)z k0 k0 k0 k0

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) + Design Methods •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