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Swept Sine Chirps for Measuring Impulse Response

Ian H. Chan

Design Engineer

Stanford Research Systems, Inc.

Log-sine chirp and variable speed chirp are two very useful test signals for

measuring frequency response and impulse response. When generating pink

spectra, these signals posses crest factors more than 6dB better than maximum-

length sequence. In addition, log-sine chirp separates distortion products from

the linear response, enabling distortion-free impulse response measurements,

and variable speed chirp offers flexibility because its frequency content can be

customized while still maintaining a low crest factor.

1. Introduction

Impulse response and, equivalently, frequency response measurements are fundamental to

characterizing any audio device or audio environment. In principle, any stimulus signal that provides

energy throughout the frequency range of interest can be used to make these measurements. In practice

however, the choice of stimulus signal has important implications for the signal-to-noise ratio (SNR),

distortion, and speed of the audio measurements. We describe two signals that are generated

synchronously with FFT analyzers (chirp signals) that offer great SNR and distortion properties. They are

the log-sine chirp and the variable speed chirp. The log-sine chirp has a naturally useful pink spectrum,

and the unusual ability to separate non-linear (distortion) responses from the linear response [1,2]. The

utility of variable speed chirp comes from its ability to reproduce an arbitrary target spectrum, all the while

maintaining a low crest factor. Because these signals mimic sines that are swept in time, they are known

generically as swept sine chirps.

2. Why Swept Sine Chirps? 1.0

Most users are probably familiar with 0.5

measuring frequency response at discrete

Amplitude (V)

frequencies. A sine signal is generated at one 0.0

frequency, the response is measured at that

frequency, and then the signal is changed to another -0.5

frequency. Such measurements have very high

signal-to-noise ratios because all the energy of the -1.0

signal at any point in time is concentrated at one 0.100 0.101 0.102

Time (s)

0.103 0.104

frequency. However, it can only manage 3

measurement rates of a handful of frequencies per

second at best. This technique is best suited to 2

making measurements where very high SNR is 1

Amplitude (V)

needed, like acoustic measurements in noisy

0

environments, or when measuring very low level

signals, like distortion or filter stop-band -1

performance. In contrast, broadband stimulus

-2

signals excite many frequencies all at once. A 32k

sample signal, for example, generated at a sample -3

0.0 0.1 0.2 0.3 0.4 0.5

rate of 64kHz can excite 16,000 different Time (s)

frequencies in only half a second. This results in Figure 1. a) Close-up of an MLS signal showing the large

excursions due to sudden transitions inherent in the signal.

much faster measurement rates, and while energy is Crest factor is about 8dB instead of the theoretical 0dB. b)

more spread out than with a sine, in many situations Three signals with pink spectra. From top to bottom, log-sine

the SNR is more than sufficient to enable good chirp, filtered MLS, and filtered Gaussian noise. The crest factor

measurements of low level signals. We will show worsens from top to bottom. All signals have a peak amplitude

of 1V. Signals offset for clarity.

several such measurements in Section 5.

page 1

SRS Inc. Swept Sine Chirps for Measuring Impulse Response

A figure of merit that distinguishes different broadband stimulus signals is the crest factor, the

ratio of the peak to RMS level of the signal. A signal with a low crest factor contains greater energy than a

high crest factor signal with the same peak amplitude, so a low crest factor is desirable. Maximum-length

sequence (MLS) theoretically fits the bill because it has a mathematical crest factor of 0dB, the lowest

crest factor possible. However, in practice, the sharp transitions and bandwidth-limited reproduction of the

signal result in a crest factor of about 8dB (Fig. 1a). Filtering MLS to obtain a more useful pink spectrum

further increases the crest factor to 11-12dB. Noise is even worse. Gaussian noise has a crest factor of

about 12dB (white spectrum), which increases to 14dB when pink-filtered.1 On the other hand, log-sine

chirp has a measured crest factor of just 4dB (Fig. 1b), and has a naturally pink spectrum. The crest

factor of variable speed chirp is similarly low, measuring 5dB for a pink target spectrum. These crest

factors are 6-8dB better than that of pink-filtered MLS. That is, MLS needs to be played more than twice

as loud as these chirps, or averaged more than four times as long at the same volume, for the same

signal-to-noise ratio.

3. Generating Swept Sine Chirps -20

Compounding the low crest factor

Power (dBVrms)

-30

advantage are the unique properties of log-sine

chirp to remove distortion, and variable speed chirp

to produce an arbitrary spectrum. To understand -40

how these properties come about requires a

knowledge of how these signals are constructed. -50

First up is the log-sine chirp. The log-sine chirp is

essentially a sine wave whose frequency increases 10 100

Frequency (Hz)

1000 10000

exponentially with time (e.g. doubles in frequency

1.0

every 10ms). This is encapsulated by [2]

2f T f2

x(t ) = sin 1 exp( ln( f1 )t ) - 1 (1)

0.5

ln( f 2 )

Amplitude (V)

f1

T

0.0

where f1 is the starting frequency, f 2 the ending

-0.5

frequency, and T the duration of the chirp. This

signal is shown in Figure 2. The explanation of its

-1.0

special property will come in Section 4, when the 0.0 0.1 0.2 0.3 0.4 0.5

signal is analyzed. Time (s)

Figure 2. a) Power spectrum of a log-sine chirp signal. It is

pink except at the lowest frequencies. b) Time record of a log-

The variable speed chirp's special property sine chirp signal. The frequency of the signal increases

comes from the simple idea of using the speed of exponentially before repeating itself.

the sweep to control the frequency response [3]. The

greater the desired response, the slower the sweep through that frequency (Fig. 3). To generate a

variable speed chirp, it is easiest to go into the frequency domain. This entails specifying both the

magnitude and phase of the signal, and then doing an inverse-FFT to obtain the desired time-domain

signal. The magnitude of a variable speed chirp is simply that of the desired target frequency response

H user ( f ) . The phase is a little trickier to specify. What we have to do is first specify the group delay of

the signal G , and then work out the phase from the group delay. The group delay for variable speed

chirp is [3]

2

G ( f ) = G ( f - df ) + C H user ( f ) (2)

where

G ( f 2 ) - G ( f1 ) . (3)

C= fS / 2

H

2

user (f)

f =0

1

True Gaussian noise has an infinite crest factor; the excursion of the noise here was limited to