CH05 Bandpass Delta-Sigma Modulation

EECE695D(POSTECH) Oversampling Analog Circuits

Bandpass Delta-Sigma Modulation

Advantage of delta-sigma modulator

Simplified anti-aliasing filtering

Inherent linearity

Robust operation

Low power consumption

Disadvantage of delta-sigma modulator

Very high sampling frequency

5.1 the need for Bandpass and Quadrature Modulation

In the RF communication system,

Want to eliminate analog components as much as possible

Want to convert into digital at as early stage as possible

Want to convert into digital at the mixer output (IF+baseband)

Difficult with the Nyquist rate ADC because of high IF freq

a bandpass delta-sigma modulator ADC suitable for this

Bandpass delta-sigma modulator ADC can perform

IF filtering

In-phase and quadrature-phase separation

High-accuracy AD conversion

Bandpass delta-sigma modulator ADC used in comm system

Sampling lowpass signal

BW=5MHz, fS >= 10MHz (Nyquist rate) to avoid aliasing

Sampling Bandpass signal (1/5)

Center frequency fC=20MHz, BW=5MHz,

fS = 17.5MHz (~ half Nyquist rate)

Understanding Digital Signal Processing: Periodic Sampling, By Richard G. Lyons

To avoid aliasing, the sampling frequency Fs must satisfy

fC+0.5BfC-0.5B-fC+0.5B-fC-0.5B

-fC+0.5B+m*fC -fC-0.5B+(m+1)*fC

-fC+0.5B+m*FS <= fC-0.5B AND -fC-0.5B+(m+1)*FS >= fC+0.5B

(2 fC + B) / (m+1) <= FS <= (2 fC - B) / mfor integer m

m * FS <= 2 fC – BW (m+1) * FS >= 2 fC + BW

(2 fC + BW) / (m+1) <= FS <= (2 fC - BW) / m

fC=20MHz, BW=5MHz

fS = (2 fC - BW) / 6

fS < (2 fC - BW) / 6

fS = (2 fC + BW) / 7

(2 fC + BW) / (m+1) <= fS <= (2 fC - BW) / m fC=20MHz, BW=5MHz

for m = 0, 45MHz <= fS <= infinity

m = 1, 22.5MHz <= fS <= 35 MHz

m = 2, 15MHz <= fS <= 17.5 MHz

m = 3, 11.25MHz <= fS <= 11.67 MHz

m = 4, 9MHz <= fS <= 8.75 MHz X

m = 5, 7.5MHz <= fS <= 7 MHz X

m = 6, 6.43MHz <= fS <= 5.83 MHz X

Allowed frequency bands: [11.25M, 11.67M], [15M, 17.5M], [22.5M, 35M], [45M,inf ]

22.5M17.5M-17.5M-22.5M

for m = 0, 45MHz <= fS <= infinity

m = 1, 22.5MHz <= fS <= 35 MHz

m = 2, 15MHz <= fS <= 17.5 MHz

m = 3, 11.25MHz <= fS <= 11.67 MHz

m = 4, 9MHz <= fS <= 8.75 MHz X

m = 5, 7.5MHz <= fS <= 7 MHz X

m = 6, 6.43MHz <= fS <= 5.83 MHz X

Bandpass delta-sigma modulator

The quantization noise stop band is centered at a non-

zero frequency, such as Fs / 4

Advantage of Bandpass delta-sigma modulator

Eliminates the analog down conversion operations

Keep the noise shaping capability

More efficient in circuit level : Narrow band is easier to

handle than wideband

(conversion range [fc-fB/2, fc+fB/2] rather than [0, fc+fB/2])

NTF poles and zeros

LP DSM: zeros at DC(z=1) or close to DC on unit circle

BP DSM: zeros at rather high frequency (eg. Fs/4, z=+- j )

Minimum sample rate at the output of decimation filter:

2 fB (Fs/OSR, LP DSM), slightly > 2fB(BP DSM)

OSR = Fs / (2 fB) defined for both LP & BP DSM

NTF for BP DSM : 1+z-2 (zeros at z=+- j)

NTF for LP DSM : 1- z-1 (zeros at z=1)

If we replace z-1 in LP DSM by –z-2 ,

we can implement a 2nd order BP DSM.

2nd order BP DSM, STF(z) = NTF(z) =

2nd order BP DSM

STF(z) = 1

NTF(z) = 1+z-2 (zeros at z=+- j (+- Fs/4))

EECE695D(POSTECH) Oversampling Analog Circuits% BPDSM2 2nd order bandpass DSM% Fs Sampling frequency % t Time vector % T Sample time % L Length of signal Fs = 1024; T = 1/Fs; L = 100 * Fs; t = (0:L-1)*T; u= 1.0 * sin (2 * pi * 266.234411 * t); v(1:L)=0.0; y(1:L)=0.0; y1_prev=0.0; y2_prev=0.0; v1_prev=0.0; v2_prev=0.0; %u1_prev=u(1); u2_prev=u(1);for i = 1: L

y(i) = u(i) - y2_prev + v2_prev;% y(i) = u2_prev - y2_prev + v2_prev;if ( y(i) > 0.0) v(i)=1; else v(i)=-1; endy2_prev=y1_prev; y1_prev=y(i); v2_prev=v1_prev; v1_prev=v(i);%u2_prev=u1_prev; u1_prev=u(i);

end%% Power spectrum density % FFT % hanning windoww=0.5*(1-cos(2*pi*(0:L-1)/L)); Ptot = (abs(fft(v(1:L).*w))).^2; Pdb = 10*log10( Ptot/max(Ptot(10:L/2))); P_avg = smooth(Pdb, 64); % subplot(1,2,1); plot(linspace(0, Fs/2, L/2), Pdb(1:L/2)); grid on; axis([0,Fs/2,-150,0]);title('Pout vs. Freq(Windowed)'); ylabel('Pout, [dB]'); xlabel('Frequency, [Hz]'); subplot(1,2,2); semilogx(linspace(0, Fs/2, L/4), Pdb(L/4:L/2-1)); grid on; axis([1e-2,512,-150,0])title('Averaged Pout (Windowed)'); ylabel('Pout, [dB]'); xlabel('Frequency-256, [Hz]');

EECE695D(POSTECH) Oversampling Analog Circuits2nd order bandpass DSM

+20dB/dec

EECE695D(POSTECH) Oversampling Analog Circuits2nd order LP DSM & 4th order BP DSM (G1=G2=1)

Design of Bandpass Delta-Sigma Modulators: R. Jacob Baker and Vishal Saxena

EECE695D(POSTECH) Oversampling Analog Circuits4th order BP DSM (G1=G2=1)

+40dB/dec

% BPDSM4 4th order bandpass DSM% Fs Sampling frequency % t Time vector % T Sample time % L Length of% signal

Fs = 1024; T = 1/Fs; L = 1000 * Fs; t = (0:L-1)*T; u= 1.0 * sin (2 * pi * 266.267314 * t); v(1:L)=0.0; y(1:L)=0.0; x(1:L)=0; x_p1=0; x_p2=0; y_p1=0; y_p2=0; v_p1=0; v_p2=0;for i = 1: L

y(i) = (x_p2-v_p2)*(-1) - y_p2;if ( y(i) > 0.0) v(i)=1; else v(i)=-1; endx(i)=u(i)-v(i)-x_p2;x_p2=x_p1; x_p1=x(i); y_p2=y_p1; y_p1=y(i); v_p2=v_p1; v_p1=v(i);

end%% Power spectrum density % FFT % hanning windoww=0.5*(1-cos(2*pi*(0:L-1)/L)); Ptot = (abs(fft(v(1:L).*w))).^2; Pdb = 10*log10( Ptot/max(Ptot(10:L/2))); P_avg = smooth(Pdb, 64); % subplot(1,2,1); plot(linspace(0, Fs/2, L/2), Pdb(1:L/2)); grid on; axis([0,Fs/2,-150,0]);title('Pout vs. Freq(Windowed)'); ylabel('Pout, [dB]'); xlabel('Frequency, [Hz]'); subplot(1,2,2); semilogx(linspace(0, Fs/2, L/4), Pdb(L/4:L/2-1)); grid on; axis([1e-2,512,-220,0])title('Pout (Windowed)'); ylabel('Pout, [dB]'); xlabel('Frequency-256, [Hz]');

EECE695D(POSTECH) Oversampling Analog CircuitsComparison: 2nd & 4th order BP DSM (G1=G2=1)

2nd: +20dB/dec,

4th: +40dB/dec

% BPDSM24 : Comparison between 2nd & 4th order bandpass DSM% Fs Sampling frequency % t Time vector % T Sample time % L Length of signal Fs = 1024; T = 1/Fs; L = 1000 * Fs; t = (0:L-1)*T; u= 1.0 * sin (2 * pi * 266.267314 * t); v(1:L)=0.0; y(1:L)=0.0; x(1:L)=0; x_p1=0; x_p2=0; y_p1=0; y_p2=0; v_p1=0; v_p2=0;for i = 1: L

y(i) = (x_p2-v_p2)*(-1) - y_p2;if ( y(i) > 0.0) v(i)=1; else v(i)=-1; endx(i)=u(i)-v(i)-x_p2;x_p2=x_p1; x_p1=x(i); y_p2=y_p1; y_p1=y(i); v_p2=v_p1; v_p1=v(i);

end%% Power spectrum density % FFT % hanning windoww=0.5*(1-cos(2*pi*(0:L-1)/L)); Ptot = (abs(fft(v(1:L).*w))).^2; Pdb4th = 10*log10( Ptot/max(Ptot(10:L/2))); P_avg4th = smooth(Pdb4th, 64); v(1:L)=0.0; y(1:L)=0.0; y1_prev=0.0; y2_prev=0.0; v1_prev=0.0; v2_prev=0.0; %u1_prev=u(1); u2_prev=u(1);for i = 1: L

y(i) = u(i) - y2_prev + v2_prev;% y(i) = u2_prev - y2_prev + v2_prev;if ( y(i) > 0.0) v(i)=1; else v(i)=-1; endy2_prev=y1_prev; y1_prev=y(i); v2_prev=v1_prev; v1_prev=v(i);%u2_prev=u1_prev; u1_prev=u(i);

endw=0.5*(1-cos(2*pi*(0:L-1)/L)); Ptot = (abs(fft(v(1:L).*w))).^2; Pdb2nd = 10*log10( Ptot/max(Ptot(10:L/2))); P_avg2nd = smooth(Pdb, 64); fp = linspace(10, Fs/2, L/2);subplot(1,2,1); plot(fp,Pdb2nd(1:L/2),'b', fp, Pdb4th(1:L/2), 'k'); grid on; axis([0,Fs/2,-220,0]);title('Pout 2nd:blue 4th:black(Windowed)'); ylabel('Pout, [dB]'); xlabel('Frequency, [Hz]'); fp = linspace(0, Fs/2, L/4);subplot(1,2,2); semilogx(fp, Pdb2nd(L/4:L/2-1), 'b',fp, Pdb4th(L/4:L/2-1), 'k' ); grid on; axis([1e-2,512,-220,0])title('Pout 2nd:blue 4th:black(Windowed)'); ylabel('Pout, [dB]'); xlabel('Frequency-256, [Hz]')

Gc: comparator gain (required)

4th order BP DSM

An example BP DSM (A GSM example)

IF frequency 10MHz,

Carrier(uplink: 933~960MHz, downnlink: 890~915MHz)

Signal band 200kHz (eg. [10M-0.1M, 10M+0.1M] )

Sampling frequency(Fs) 40MHz

OSR = 40MHz / (2 * 200kHz) = 100

A 3rd order binary LP DSM with OSR=100 SQNR=100dB

A 6th order BP DSM with OSR=100 the same SQNR

The lowest frequency that can be aliased into signal band

39.8MHz in LP DSM 199x fB trivial to design an anti-aliasing filter

29.9MHz in BP DSM 3x max freq needs a BPF for anti-aliasing,

Anti-aliasing filter not necessary with the continuous time loop filter (NO SAMPLING)27

5.2 Bandpass NTF selectionFo: center frequency, Fs: sampling frequency, Fb; signal bandwidth

(1) If Fo & Fb : fixed Large Fs preferred (Large OSR Large SQNR)

easy to design an anti-aliasing filter between Fo & Fs

(2) If Fb is fixed small Fo preferred (easy anti-aliasing, improve noise

shaping, but the low frequency noise limits min Fo,

(3) Fo=Fs/4(a rational) simplifies the decimation filter design

6th order Bandpass DSMFo = Fs / 6 , OSR=32 (Fb = Fs/32), NTF out-of-band gain = 2

5.2.1 Pseudo N-path transformation

Transformation(1) z -z : LPF HPF, HPFLPF

TT)( TwjTwjTj eee

5.0 5.0 5.10 2

z -z11

1 ze aT HPF

1 ze aT

Transfer function Impulse response Impulse response Transfer function

Transformation(1) z -z2 : pseudo 2-path transformation

T 22)22( mTwjTj ee

Integrator

mag )cos(

T resonator

LPF BPF, HPF BRF, Integrator resonator

Transformation

(1) z z2 : 2-path transformation

T 2)22( mTwjTj ee

1 ze aT

5.2.1 Pseudo N-path transformationz -z2 : order-n highpass NTF order-2n band-reject NTF

n zeros near z=1 n zeros near z=j

n zeros near z=-j

z -z2 : NTF1(z)=(1-z-1)n NTF2(z)=(1+z-2)n

n zeros near z=1 n zeros near z=j

n zeros near z=-j

NTF2(z) has the same gain and frequency profile as NTF1(z),

but compressed by a factor of 2 in frequency and replicated.

2 - path

Polyphase filter to implement BP DSM

STF(z) = 1

NTF(z) = 1 + z-2

| NTF | = 2 | cos(wT) | BRF 40

EECE630, K.J.Ray Liu, Min Wu, University of Maryland, College Park

Polyphase implementation: lowers the ckt operating speed

5.2.1 Pseudo 2-path transformation

Two methods for the pseudo 2-path transformation

(1) Apply z -z2 transform to H(z), and then synthesize

the resultant transfer function H’(z)

(2) Replace a delay block(z-1) in circuit by a series

connection of two delay blocks and an inversion

(-z-2)

Apply z -z2 to MOD1(2) Replace a delay block(z-1) in circuit by a series connection of two delay blocks and an

inversion (-z-2)

Polyphase filter to implement BP DSM

5.3 Architecture for bandpass delta-sigma modulator

Low-pass DSM

If output taken from x1 & x3 instead of from x2 & x4

LP DSM BP DSM

______

IEEE JSSC vol.32, NO. 12, DECEMBER 1997, pp.1935, Quadrature Bandpass DSM for Digital Radio, Stephen A. Jantzi et al.

Bandpass DSM

Resonator: Assume g1=1

5.3.1 Topology choices

Loop filter for BP DSM

4 x DAC(a1, a2, a3, a4)

)()( 220

BPF BRF

3 )()()()( aaasaasassA 54

5.3.2 Resonator implementations

(1) Bandpass DSM needs resonators, a LP DSM needs integrators

(2) Finite quality factor(Q) of resonator degrades performance, as the

finite OP amp gain degrades performance in LP DSM

(3) Q must be high (Q >> f0 / fb)

5.3.2 Resonator implementations

Switched capacitor implementation of resonator using LDI loop

( LDI: lossless discrete integrator)

Resonator implemented with LDI loop

Poles of characteristic equation: on the unit circle

Q is infinite (limited by OP amp gain)

Q > 100 easily obtained)5.01(cos 1 gp 59

Is this LDI ?

DDI versus LDI (1/2)EECE695D(POSTECH) Oversampling Analog Circuits

U.C.Berkeley 61

U.C.Berkeley

DDI versus LDI (2/2)

U.C.Berkeley 62

Gm-C resonator

LG = - {Gm / (sC) }2

Vout / Vin = +(Gm1/sC)*(Gm/(sC) / [ 1 + {Gm / (sC) }2

= {( Gm1 * Gm ) / C2} / { s2 + wo2 }

Resonance frequency wo = Gm/C

wo 30% variableneeds tuning

Finite Rout of amp &

delay of transconductor

limits Q

Active RC resonatorResonance frequency wo = 1/(RC)

wo variableneeds tuning

LG = -1/(sRC)2

LC resonatorResonance frequency wo = 1/sqrt(LC)

On-chip inductor : a few nH must operate at multi GHz

Discrete inductor : tolerance 2% Q > 50

Discrete capacitor: tighter tolerance, larger Q than inductor

LC tank low noise, low distortion, low power

Fig. 5-25 : NTF at signal band > 50dB

IF 273MHz, LO 269 MHz, CLK 32MHz

Fig. 5-25 :

CH05 Bandpass Delta-Sigma Modulation

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