# 0.2 Measuring impulse responses using golay complementary sequences

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Two impulse-response measurement methods are demonstrated, with source code provided in Matlab and Pure Data (PD).The measurement technique using Golay complementary sequences is particularly robust to additive white noise, while the swept sine measurementtechnique is robust to a weakly nonlinear motor exciting the linear system being measured.

## Introduction

[link] depicts a linear system characterized by an impulse response $h\left(n\right)$ , driven by an input signal $s\left(n\right)$ , and producing the output signal $r\left(n\right)$ . The system identification problem is to estimate $h\left(n\right)$ given known input/output signals $s\left(n\right)$ and $r\left(n\right)$ . This module illustrates the swept-sine and the Golay complementary sequence techniques for identifying finite impulse responses. The firstexample measures the impulse response of a highpass filter using the Golay method, and the second example measures the impulse response ofa weakly nonlinear loudspeaker driver using the swept-sine method.

## Summary of objectives

• To provide a general framework for characterizing single-input, single-output linear systems .
• To demonstrate how to measure the impulse response of a linear system using Golay complementary sequences .
• To explain some of the limitations of making measurements using standard sound interfaces.
• To demonstrate how to find the minimum-phase spectrum corresponding to a complex spectrum.
• To explain how to measure the impulse response of a system even if the excitation source motor is weakly nonlinear . This measurement technique uses a sine sweep test signal.

## Installation

1. Install a sound card, sound interface, or other full-duplex data acquisition card.
2. Consider testing the data acquisition card viewing the soundcard set-up instructions .
3. Install either MATLAB or Octave .
4. Install pd . (Alternatively, you may use other software that is capable of recording a system's output for a given inputexcitation signal. The software should also be capable of reading and writing WAV files. For example, any multitrack recordingsoftware should be fine.)
5. Download tf_meas.zip and unzip the contents into a conveniently located local directory.

## Linear system

Consider the causal, single-input single-output (SISO) system shown in [link] . For simplicity, we will take the system to be linear and discrete-time, so that it is characterized by its impulseresponse $h\left(n\right)$ or equivalently its transfer function $H\left(z\right)$ , which is the $z$ transform of $h\left(n\right)$ .

$h\left(n\right)↔H\left(z\right)$

We will assume that both $h\left(n\right)$ and $H\left(z\right)$ exist so that we can discuss measuring them interchangeably. We will further assume that $h\left(n\right)$ has finite length so that we can measure the response to an input signal in a finite amount of time. The goal of this document isto explain how to excite the system with a signal $s\left(n\right)$ , measure the response $r\left(n\right)$ , and use $s\left(n\right)$ and $r\left(n\right)$ to determine $h\left(n\right)$ (and equivalently $H\left(z\right)$ ). In particular, it is useful to pick a signal $s\left(n\right)$ that contains a large amount of energy so that measurement noise will not significantly corrupt the measurement results.

## Limitations of sound interfaces

Sound cards and sound interfaces are not designed for making transfer function measurements. They merely provide a cost-effective solutionsince almost all computers have sound cards. We demonstrate these weaknesses given measurements made on a PreSonus Firepod soundinterface. The output from channel 1 was directly connected to the line input on channel 1, and the sampling rate was ${f}_{S}=44.1$ kHz.

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