Analog Integrated Circuits (Jieh Tsorng Wu)

Requirements for the analog circuitry are less severe

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Unformatted text preview: is very important. Oversampling 26-9 Analog ICs; Jieh-Tsorng Wu Second-Order ∆Σ Modulator x(k) z u z 1 u 2 y(k) 1 1 D/A The transfer functions are ST F (z ) = Y (z ) = z −1 X (z ) NT F (z ) = Y (z ) = 1 − z −1 E (z ) 2 The noise transfer function in frequency domain is πf |NT F (f )| = 2 sin fs Oversampling 26-10 2 Analog ICs; Jieh-Tsorng Wu Second-Order ∆Σ Modulator If OSR 1, the quantization noise power is 24 ∆π 1 Pn ≈ 60 OSR5 And SNRy,max = 15 × 22N × OSR5 = −11.14 + 6.02 · N + 50 log(OSR) dB 2π4 • Oversampling gives a SNR improvement of 15 dB/octave or 2.5 bit/octave. Oversampling 26-11 Analog ICs; Jieh-Tsorng Wu Integration Range in a Second-Order ∆Σ Modulator The outputs of the integrators are u1(k + 1) = x (k ) − e(k ) + e(k − 1) u2(k + 1) = x (k − 1) − 2e(k − 1) + e(k − 2) For multi-bit quantization: • For small |x |, e is bounded by ±∆/2. • If |e| < ∆/2, then |u1| ≤ |x (k )| + |e(k )| + |e(k − 1)| ≤ |x | + ∆ 3 |u2| ≤ |x (k )| + 2|e(k = 1)| + |e(k − 2)| ≤ |x | + ∆ 2 Oversampling 26-12 Analog ICs; Jieh-Tsorng Wu Integration Range in a Second-Order ∆Σ Modulator • One-bit quantization. • D/A output levels are ±1. • The integrators are bounded by |u1|max = |x | + 2 2 |u2|max = (5 − |x |) 8(1 − |x |) • In practice, the first accumulation is often clipped at ±2, and the second effectively ±4. Oversampling 26-13 Analog ICs; Jieh-Tsorng Wu Overloading in a Second-Order ∆Σ Modulator • D/A levels are ±0.5, ±1.5, and ±2.5. • ∆ is the same for all three cases. • For large x , the input to the quantizer can be so large that |e| > ∆/2. The excess noise can degrade the SNR of y . • In the two-level case (1-bit quantization), the comparator is theoretically overloaded for all conditions, except zero input with zero initial conditions. Oversampling 26-14 Analog ICs; Jieh-Tsorng Wu Oversampling ADCs x c (t) Anti− Aliasing Filter x(n) Sampling y(n) ∆Σ yp (n) y (n) b Digital Low−Pass Filter Modulator L Decimation Filter fs Xc(j Ω) x c (t) Ω t 0 2Ts 6Ts 4Ts 0 Ωb Ωs X ej ω x(n) n 0123456 Oversampling 0 26-15 2π ω Analog ICs; Jieh-Tsorng Wu Oversampling ADCs Ye y(n) jω n 2π 0 Yp e yp (n) jω n 0123456 2π 0 Yb e y (n) b Oversampling 1 2 3 0 26-16 ω jω n 0 ω 2π 4π ω Analog ICs; Jieh-Tsorng Wu Oversampling DACs x b (n) x p (t) x(n) y(n) Interpolation Low−Pass Filter L ∆Σ D/A Modulator fs Xb e x (n) b y (t) d y (t) c Digital Analog Filter jω n 0 1 2 3 2π 0 X p ej ω x (n) p n 0123456 Oversampling ω 0 26-17 2π 4π ω Analog ICs; Jieh-Tsorng Wu Oversampling DACs Xe x(n) jω n 0123456 2π 0 y(n) Ye ω jω n 2π 0 y (t) d ω Yd (j Ω) t Ω 0 Oversampling 0 2Ts 4Ts 6Ts 26-18 Ωb Ωs Analog ICs; Jieh-Tsorng Wu Oversampling DACs y(n) Ye jω n 2π 0 y (t) d ω Yd (j Ω) t Ω 0 0 2Ts 4Ts 6Ts Ωb Ωs Yc(j Ω) y c (t) Ω t 0 Oversampling 0 26-19 Ωb Ωs Analog ICs; Jieh-Tsorng Wu General Single-Stage ∆Σ Modulator x(k) y(k) G(z) F(z) Y (z ) = G (z ) 1 · X (z ) + · E (z ) = ST F (z ) · X (z ) + NT F (z ) · E (z ) 1 + F (z )G (z ) 1 + F (z )G (z ) • OSR is typically between 16 and 256. • The loop gain, L(z ) = F (z )G (z ), need to be high in the band of interest. • The poles L(z ) are the zeros of NT F (z ). • Both ST F (z ) and NT F (z ) generally share the same poles, the roots of 1 + L(z ) = 0. Oversampling 26-20 Analog ICs; Jieh-Tsorng Wu General Single-Stage Error-Feedback Coder y(k) x(k) N(z) − 1 e(k) Y (z ) = X (z ) + N (z ) · E (z ) • A slight coefficient error can degrade noise-shaping significantly. • Not suitable for analog modulators, only appropriate for digital modulators. Oversampling 26-21 Analog ICs; Jieh-Tsorng Wu Single-Stage High-Order Modulators c1 z x(k) 1 1 z z 1 1 a1 c2 1 z z 1 1 a2 1 z z 1 1 a3 1 z z 1 1 a4 1 z 1 a5 y(k) • An N th-order noise-shaping modulator improves the SNR by (6N + 3) dB/octave, or equivalently, (N + 0.5) bits/octave. Oversampling 26-22 Analog ICs; Jieh-Tsorng Wu Single-Stage High-Order Modulators If c1 = c2 = 0, L(z ) = G (z ) = a1 (z − + 1)1 a2 (z − 1)2 + a3 (z − 1)3 + ··· n (z − 1) 1 = NT F (z ) = 1 + L(z ) D (z ) ST F (z ) = 1 − NT F (z ) • L(z ) has all its poles at z = 1 (or f = 0). • NT F (z ) has all its zeros at z = 1 (or f = 0). • Butterworth high-pass filters are often used for NT F (z ). • ST F (z ) contains peaking at high frequencies. If c1 = 0 and c2 = 0, the poles of L(z ) can be moved away from z = 1 along the unit circle. Oversampling 26-23 Analog ICs; Jieh-Tsorng Wu Single-Stage High-Order Modulators x(k) b1 b2 b3 b4 b5 c1 z 1 a1 1 z z 1 1 a2 c2 1 z z 1 1 a3 1 z z 1 1 a4 1 z z 1 1 1 z 1 a5 y(k) Oversampling 26-24 Analog ICs; Jieh-Tsorng Wu Single-Stage High-Order Modulators If c1 = c2 = 0, L(z ) = G (z ) = a1 1)n−0 (z − b1 1)n−0 (z − + + a2 1)n−1 (z − b2 (z − 1)n−1 + + 1)n−2 (z − b3 (z − 1)n−2 + ··· + ··· 2 n (z − 1) 1 NT F (z ) = = 1 + L(z ) D (z ) a3 ST F...
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This note was uploaded on 03/26/2013 for the course EE 260 taught by Professor Choma during the Winter '09 term at USC.

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