Skip to content Skip to navigation

Connexions

You are here: Home » Content » Golay Code Impulse Response Measurement

Navigation

Content Actions
  • Download module PDF
  • Add to ...
    Add the module to:
    • My Favorites
    • A lens
    • An external social bookmarking service
    • My Favorites (What is 'My Favorites'?)
      'My Favorites' is a special kind of lens which you can use to bookmark modules and collections directly in Connexions. 'My Favorites' can only be seen by you, and collections saved in 'My Favorites' can remember the last module you were on. You need a Connexions account to use 'My Favorites'.
    • A lens (What is a lens?)

      Definition of a lens

      Lenses

      A lens is a custom view of Connexions content. You can think of it as a fancy kind of list that will let you see Connexions through the eyes of organizations and people you trust.

      What is in a lens?

      Lens makers point to Connexions materials (modules and collections), creating a guide that includes their own comments and descriptive tags about the content.

      Who can create a lens?

      Any individual Connexions member, a community, or a respected organization.

    • External bookmarks
  • E-mail the authors

Recently Viewed

Golay Code Impulse Response Measurement

Module by: Edgar Berdahl, Julius Smith

Summary: The Golay code measurement technique is used for measuring impulse responses. Compared to the swept-sine method, it is more robust to additive white noise. Free, open-source software is provided in Matlab (or Octave) and Pure Data (PD) for carrying out Golay-code impulse-response measurements using the sound hardware found on a typical personal-computer.

Introduction

Figure 1: Linear system to be measured
linsystem2.png

Figure 1 depicts a linear system characterized by an impulse response h(n)h(n), driven by an input signal s(n)s(n), and producing the output signal r(n)r(n). The system identification problem is to estimate h(n)h(n) given known input/output signals s(n)s(n) and r(n)r(n). A practical method for identifying finite impulse responses is the Golay-code measurement technique, described below. Maximum length sequences may be alternatively used for this type of measurement, as they are also noise signals consisting of 1's and -1's, but perfectly inverse filtering the measurement is more computationally intensive.

Golay Code Theory

The length LL bilevel sequences a(n)a(n) and b(n)b(n) are Golay if and only if the following condition holds, where ¤¤ denotes the autocorrelation operator Entry 1:

a ( n ) ¤ a ( n ) + b ( n ) ¤ b ( n ) = 2 L δ ( n ) a ( n ) ¤ a ( n ) + b ( n ) ¤ b ( n ) = 2 L δ ( n ) (1)

and δ(n)δ(n) is the Kronecker delta function. Recall that (Equation 1) can also be written using **, the convolution operator:

a ( - n ) * a ( n ) + b ( - n ) * b ( n ) = 2 L δ ( n ) a ( - n ) * a ( n ) + b ( - n ) * b ( n ) = 2 L δ ( n ) (2)

Given that aL(n)aL(n) and bL(n)bL(n) are Golay, it turns out that a2L(n)=[aL(n)bL(n)]a2L(n)=[aL(n)bL(n)] and b2L(n)=[aL(n)-bL(n)]b2L(n)=[aL(n)-bL(n)] are also Golay. This means that Golay sequences can be constructed recursively given Golay seed sequences such as a2(n)=[11]a2(n)=[11] and b2(n)=[1-1]b2(n)=[1-1]. See the MATLAB/Octave source code generate_golay.m for details. Notice also that the resulting bilevel sequences consist of only 1's and -1-1's. This means that the signal contains the maximum possible power level given that |s(n)|≤1∀n|s(n)|≤1∀n. This property is helpful for minimizing measurement noise.

Let ra(n)=a(n)*h(n)ra(n)=a(n)*h(n) be the response due to the Golay code input a(n)a(n), and let rb(n)=b(n)*h(n)rb(n)=b(n)*h(n) be the response due to the Golay code input b(n)b(n). Due to (Equation 1), the impulse response h(n)h(n) may be determined as follows:

h ( n ) = 1 2 L ( a ( n ) ¤ r a ( n ) + b ( n ) ¤ r b ( n ) ) h ( n ) = 1 2 L ( a ( n ) ¤ r a ( n ) + b ( n ) ¤ r b ( n ) ) (3)

See golay_response.m for more details.

Golay Code Measurement Procedure and Software

  1. Generate the Golay codes golayA.wav and golayB.wav using generate_golay.m.
  2. Open the pd patch golay.pd, in pd.
  3. Ensure that the patch is not in editing mode, and check the “compute audio” box in the main pd window.
  4. Adjust the “Output Volume” so that when you click on “Record Response to Golay A”, the system under test is behaving linearly (i.e. not clipping), but so that the input signal to the sound interface is not too noisy.
  5. If there is an input volume on the sound interface, adjust it so that the levels approximately match those shown in Figure 2 when you click on “Record Response to Golay A” and “Record Response to Golay B.” If the sound interface has no input volume, then you will need to adjust the “Output Volume” accordingly.
    Figure 2: golay.pd after making a measurement with an appropriate input level
    golayScale.png
  6. Once you are satisfied with the results, click the “Write Responses to Disk” button.
  7. pd will write the files RespA.wav and RespB.wav to disk. Rename these files so that the names match the measurement you just made. For instance, you might rename them to hpfRespA.wav and hpfRespB.wav if they corresponded to the measurement of a high-pass filter.
  8. Run golay_response('hpf') in MATLAB or Octave to analyze the measured response. Plots will be created, and the file hpfImpResp.wav will be written to disk.

References

  1. Abel, J. and Berners, D. (2005). Signal Processing Techniques for Digital Audio Effects. http://ccrma.stanford.edu/courses/424/.

Comments, questions, feedback, criticisms?

Send feedback