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Design and Application of a Hilbert Transformer in a Digital Receiver SDR11 - WInnComm November 29, 2011 Matt Carrick Motivation Given real spectrum at arbitrary IF, how to get to complex baseband? Constraints: Real A/D

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Design and Application of a Hilbert Transformer in a Digital Receiver

November 29, 2011 Matt Carrick

SDR’11 - WInnComm

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Motivation

  • Given real spectrum at arbitrary IF, how to get to complex baseband?
  • Constraints:

– Real A/D – Minimize Processing Power

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Outline

  • Comparison of quadrature downconverter, downconversion

with Hilbert transformer

  • Hilbert Transform Review
  • Hilbert Transform Filter Design Through Windowing
  • Hilbert Transform Filter Design in Frequency
  • Designing a Half Band Filter
  • Results
  • Implementation of Hilbert Transform Filter

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Downconversion Options

  • Quadrature Downconverter
  • Hilbert Transform + Heterodyne
  • Other Options

– Alias to baseband, Polyphase filter bank + FFT

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Downconversion With Quadrature Downconverter

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Downconversion With Hilbert Transform

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Hilbert Transform Review

  • Convolutional Operator, Analog Representation

– x’(t) = x(t) * h(t) – h(t) = 1/πt

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Hilbert Transform Review (Con't)

  • Digital Representation

– x’[n] = x[n] * h[n] – h[n] = 2/(πn) for n odd – h[n] = 0 for n even

  • Hilbert Transform
  • Hilbert Transformer

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Outline

  • Comparison of quadrature downconverter, downconversion with

Hilbert transformer

  • Hilbert Transform Review
  • Hilbert Transform Filter Design Through Windowing
  • Hilbert Transform Filter Design in Frequency
  • Designing a Half Band Filter
  • Results
  • Implementation of Hilbert Transform Filter

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Building Filter from Discrete Sequence

  • h[n] = 2/(πn) for n even
  • h[n] = 0 for n odd

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Reducing Ripple

  • Ripple due to Gibbs’ Phenomenon
  • Window coefficients to combat ripple

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Reducing Ripple (Con't)

  • Force tails of filter to zero artificially through windowing

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Reducing Ripple (Con't)

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Change Design Method

  • Choosing ‘best’ window is difficult
  • Instead of designing in time, design in frequency

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Outline

  • Comparison of quadrature downconverter, downconversion with

Hilbert transformer

  • Hilbert Transform Review
  • Hilbert Transform Filter Design Through Windowing
  • Hilbert Transform Filter Design in Frequency
  • Designing a Half Band Filter
  • Results
  • Implementation of Hilbert Transform Filter

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Hilbert Transform Frequency Response

  • By definition:

– H(w) = -j sgn (w) – Approximate with two half band filters

  • How to build a half band filter?

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Half Band Filter Design

  • A half band filter, filters half the spectrum
  • Every other coefficient is zero
  • Quick design method (MATLAB code);

– f = [0 wc 1‐wc 1]; – a = [1 1 0 0]; – hb = firpm(N‐1,f,a);

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Half Band Filter Design (Con't)

  • Coefficients have ‘zeros’ every other sample
  • Frequency response covers appropriate band

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Half Band Filter Design (Con't)

  • Parks-McClellan doesn’t set zero coefficients to exactly zero
  • Force coefficients to zero

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Half Band Filter Design (Con't)

  • Change in frequency response is negligible

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Sum Half Band Filters

  • G(θ) = HB(θ – π/2)
  • G(-θ) = - HB(θ + π/2)
  • HHT(θ) = j ( G(θ) + G(-θ) )
  • HHT(θ) = j ( HHB(θ – π/2) - HHB(θ + π/2) )

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Sum Half Band Filters (Con't)

  • HHT(θ) = j ( HHB(θ – π/2) - HHB(θ + π/2) )
  • HHB(θ – π/2) ↔ j hHB[n]exp(-jπn/2)
  • HHB(w + π/2) ↔ j hHB[n]exp(jπn/2)
  • hHT[n] = 2 hHB[n] sin(πn/2)

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Hilbert Transform Coefs from Half Band Coefs

  • hHT[n] = 2 hHB[n] sin(πn/2)

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Hilbert Transform Coefs from Half Band (Con't)

  • hHT[n] = 2 hHB[n] sin(πn/2)

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Hilbert Transform Coefs from Half Band (Con't)

  • Sine wave applies Hilbert transform filter properties

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Hilbert Transform Filter Response

  • Greatly improved passband ripple

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Outline

  • Comparison of quadrature downconverter, downconversion with

Hilbert transformer

  • Hilbert Transform Review
  • Hilbert Transform Filter Design Through Windowing
  • Hilbert Transform Filter Design in Frequency
  • Designing a Half Band Filter
  • Results
  • Implementation of Hilbert Transform Filter

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Implementation

  • Hilbert Transformer operates on imaginary portion
  • Delay real portion accordingly

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Total Computations Required

  • Quadrature Downconverter

– Two low pass filters of order N, two multiplies

  • Downconversion with Hilbert Transformer

– One filter of order N, one complex multiply

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Conclusion

  • Compared quadrature downconverter and downconversion with

Hilbert transformer

  • Reviewed Hilbert transform
  • Discussed windowing Hilbert transform filter
  • Designed Hilbert transform filter in frequency
  • Covered Design process for half band filter
  • Results
  • Implementation

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