Oversampling Converters

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  • IsLab |9&hr%7 z @/ Analog Integrated Circuit Design OSC-41'

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    Oversampling Converters

    _

    >

    Kyungpook National University

    Integrated Systems Lab, Kyungpook National University

    IsLab |9&hr%7 z @/ Analog Integrated Circuit Design OSC-1'

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    Oversampling Converters

    Recently popular for high-resolution medium-to-low-speed

    applications such as high-quality digital audio.

    Reduced requirements on the analog circuitry at the expense of more

    complicated high-speed digital circuitry.

    Only a first-order antialiasing filter is required for A/D converters.

    A S/H is usually not required at the input of an oversampling A/D

    converter with a SC modulator.

    Extra resolution by sampling much faster than the Nyquist rate.

    Extra resolution in lower oversampling rates by spectrally shaping the

    quantization noise ( modulation).

    Integrated Systems Lab, Kyungpook National University

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    Oversampling without Noise Shaping

    A linear model of quantizers.

    x

    y

    x(n) y(n) x(n) y(n)

    e(n)

    y(n) = x(n) + e(n)

    Quantization noise (error) modeling: white noise approximation for

    very active x(n) a random number uniformly distributed betweenVLSB/2 (VLSB ), normalized noise power of V 2LSB/12

    S2ndf ,

    root spectral density Sn(f) = VLSB/

    12fs for |f | fs/2 (two-sideddefinition of noise power) or 0 f fs (one-sided definition).

    Integrated Systems Lab, Kyungpook National University

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    Oversampling Advantage

    Oversampling ratio: Nyquist rate = 2f0, OSR fs/2f0.

    Maximum SNR: ratio of maximum sinusoidal power to quantization

    noise, signal amplitude = Vref/2 =2N/2.

    Ps =

    (

    2N

    2

    2

    )2

    =222N

    8, Pn =

    fs/2

    fs/2

    S2n(f)|H(f)|2df =1

    OSR

    2

    12

    SNRmax = 10 log(Ps/Pn) = 6.02N + 1.76 + 10 log(OSR)

    SNR improvement 3 dB/octave 0.5 bit/octave

    x(n) y1(n)H(f)

    y(n)

    f0f0 fs/2fs/2

    |H(f)|1.0

    Integrated Systems Lab, Kyungpook National University

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    If eight samples of a signal are averaged, this low-pass filtering results

    in the OSR being approximately equal to 8.

    SNR improvement = 10 log(8) = 9 dB

    For a 1-bit A/D converter with 6-dB SNR, what is fs required using

    oversampling to obtain a 96-dB SNR if f0 = 25 kHz?

    fs 2(966)/3 2f0 54, 000 GHz (33, 187 GHz)

    Noise shaping is needed to improve the SNR more faster.

    Oversampling does not improve linearity. If a 16-bit ADC is designed

    using a 12-bit converter with oversampling, the 12-bit converter must

    have an integral nonlinearity error better than 16-bit accuracy (1/216

    = 0.0015 %). 1-bit D/A converters with only two values which define

    a straight line have inherent linearity. realization of 16 to 20-bitlinear ADCs using noise shaping without trimming

    Integrated Systems Lab, Kyungpook National University

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    Oversampling with Noise Shaping

    The system architecture of a oversampling A/D converter.

    Antialiasing filter: fc fs/2, a simple RC LPF for large OSR. Delta-sigma modulator : converts the analog signal into a noise-shaped

    low-resolution digital signal. For a SC modulator, a separate S/H

    is not required.

    Decimator : converts the oversampled low-resolution digital signal

    into a high-resolution digital signal at a lower sampling rate.

    xi(t)Anti-

    aliasing

    filter

    xc(t)

    fc

    Sample

    and

    hold

    xs(t)

    fs

    mod

    xd(n)

    fs

    Digital

    low-pass

    filter

    xl(n)

    fs

    Down

    sampler

    (OSR)

    xo(n)

    2f0

    Decimation filter

    Analog Digital

    Integrated Systems Lab, Kyungpook National University

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    Noise-Shaped Delta-Sigma Modulator

    Transfer function of a delta-sigma modulator: For noise shaping, the

    magnitude of H(z) is large from 0 to f0, but that of N(z) is small.

    Y (z) =H(z)U(z)

    1 + H(z)+

    E(z)

    1 + H(z) S(z)U(z) + N(z)E(z)

    u(n) H(z)

    x(n) y(n)

    Quantizer

    delta

    sigma

    u(n) H(z)

    x(n) y(n)e(n)

    Linear model

    Integrated Systems Lab, Kyungpook National University

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    First-Order Noise Shaping

    The z-transform and the Laplace transform: two-sided definitions (z esT )

    X(z)

    X

    n=

    x(n)zn, X(s)

    Z

    x(t)estdt

    Unit delay H(z) = z1: k = 1.

    X

    n=

    x(n k)zn =

    X

    n=

    x(m)zmzk = X(z)zk

    The noise transfer function N(z) should have a zero (a pole of H(z)) at dc

    (z = esT = 1) high-pass filtering for noise.

    A discrete-time integrator : low-pass filter, accumulator.

    H(z) =1

    z 1=

    z1

    1 z1 y(n) = y(n 1) + x(n 1)

    Integrated Systems Lab, Kyungpook National University

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    Time Domain View

    Since the integrator has infinite dc gain, the average value of

    u(n) y(n) equals zero. y(n) = u(n 1) + e(n) e(n 1) Example: x(0) = 0.1, 1.0 quantizer with threshold at zero.

    n u(n) x(n) y(n) e(n) x(n + 1)

    0 1/3 0.1 1.0 0.9 0.5667

    1 1/3 0.5667 1.0 0.4333 0.7667

    2 1/3 0.7667 1.0 0.2333 0.1

    u(n) z1 y(n)x(n)

    Quantizer

    x(n + 1)

    Delay

    Integrated Systems Lab, Kyungpook National University

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    Frequency Domain View

    Signal and noise transfer functions: delay, differentiator (HPF).

    S(z) =H(z)

    1 + H(z)= z1, N(z) =

    1

    1 + H(z)= 1 z1

    Frequency response for noise: z = ejT = ej2f/fs.

    N(f) = 1 ej2f/fs = 2jejf/fs sin(

    f

    fs

    )

    Maximum SNR: sin(f/fs) f/fs for f0 fs.

    Pn =

    f0

    f0

    S2n(f)|N(f)|2df 22

    36

    (

    1

    OSR

    )3

    SNRmax = 6.02N + 1.76 5.17 + 30 log(OSR) 1.5 bits/octave

    Integrated Systems Lab, Kyungpook National University

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    Realization of A First-Order Modulator

    First-order modulator and SC implementation.

    u(n) z1 +

    y(n)

    1-bit D/A

    Quantizer

    vi

    1 C 2

    +

    C

    +

    Comparator

    vo12

    VR2

    VR2

    Latch on 2 falling

    vd

    vx

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    Second-Order Noise Shaping

    Transfer function: linear quantizer model.

    U Y + V z1 = V, (V Y + W )z1 = W

    Y (z) = W + E = z1U(z) + (1 z1)2E(z)

    S(z) = z1, N(z) = (1 z1)2

    Second-order modulator.

    u(n) z1

    y(n)

    z1 Quantizer

    v w

    Integrated Systems Lab, Kyungpook National University

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    Maximum SNR: z = ejT = ej2f/fs, sin(f/fs) f/fs for f0 fs.

    |N(f)| =[

    2 sin

    (

    f

    fs

    )]2

    , Pn 24

    60

    (

    1

    OSR

    )5

    SNRmax = 6.02N + 1.76 12.9 + 50 log(OSR) 2.5 bits/octave

    Noise transfer-function curves: Pn in the signal band (0 to f0).

    f

    |N(f)|

    0 f0 fs/2 fs

    second order

    first order

    no noise shaping

    Integrated Systems Lab, Kyungpook National University

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    Error-Feedback Structure

    Error-feedback structure of a general modulator.

    u(n) y(n)

    G(z) 1

    e(n)

    x(n)

    Transfer function for a first-order modulator.

    Y (z) = U(z) + G(z)E(z), S(z) = 1, N(z) = G(z) = 1 z1

    Integrated Systems Lab, Kyungpook National University

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    Mismatches of the analog subtracters significant noise-shapingdegradation. For example, if the subtraction becomes e = y 0.99xrather than e = y x, then N(z) = 1 0.99z1, and the zero ismoved off dc. The noise is not fully nulled at dc. Well suited todigital implementations where no coefficient mismatches occur.

    Error-feedback structure of a 2nd-order : G(z) = (1 z1)2.

    u(n) y(n)

    z1

    z1 2

    e(n)

    x(n)

    Integrated Systems Lab, Kyungpook National University

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    System Architecture of Delta-Sigma ADCs

    System architecture for a typical oversampling ADC.

    xi(t)Anti-

    aliasing

    filter

    xc(t)

    fc

    Sample

    and

    hold

    xs(t)

    fs

    mod

    xd(n)

    fs

    Digital

    low-pass

    filter

    xl(n)

    fs

    Down

    sampler

    (OSR)

    xo(n)

    2f0

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