2005 IEEE Workshop on Applications of Signal Processing to Audio and Acoustics October 16-19, 2005, New Paltz, NY SPECTRAL, BISPECTRAL, AND DUAL-FREQUENCY ANALYSIS OF TUBE AMPLIFIED ELECTRIC GUITAR SOUND Alfred Hanssen and Tor Arne Øigård Dept. of Physics, University of Tromsø NO–9037 Tromsø, Norway {alfred,torarne}@phys.uit.no Yngve Birkelund Tromsø University College NO–9293 Tromsø, Norway yngve.birkelund@hitos.no ABSTRACT We have analyzed the sound of an electric guitar that has been amplified by a high-quality all-tube amplifier, and emitted by means of a speaker cabinet. We re-amplified a recording of a clean gui- tar through a state-of-the-art all-tube amplifier at three different preamplifier gain settings: one clean, one half-distorted, and one massively distorted. Spectral analysis of recordings of the three signals exhibited a remarkably rich overtone spectum, and we ob- served that only the high frequency part of the spectrum was boosted by an increase in the distortion levels. A bispectral analysis of the amplified guitar sound showed that quadratic nonlinearities are re- sponsible for coherent phase coupling among the partials, and that the fraction of the total power which is due to quadratic nonlin- earities is larger for the clean sound than for the distorted sound. Finally, a dual-frequency analysis showed that the sound, even for the sustained part of a single string pluck, is in fact a nonstationary random process. Our analysis showed that the guitar tone should be be classified as an (almost) cyclostationary random process. 1. INTRODUCTION Discriminating professional guitar players almost invariably pre- fer all-tube guitar amplifiers over solid-state amplifiers [1, 2]. The reason for this preference is that the nonlinear behavior of the vac- uum tubes generates a harmonic overtone spectrum that is subjec- tively described as “musical”, “pleasant”, “natural”, “dynamic”, and “organic” [3]. Still, the overall response of the full guitar — tube amplifier — loudspeaker system is poorly understood. This becomes evident in listening tests of digitally simulated tube am- plifiers, where most, if not all, digital simulations sound “one- dimensional” and lifeless compared to real tube amplifiers. In par- ticular, guitarists report that digital tube amplifier simulations lack the playing feel and response of a real tube amplifier. Armed with advanced techniques from statistical analysis and modeling of random processes, we aim at shedding some light on the properties of tube-driven gitar tones. In particular, we have conducted an experiment where a clean electric guitar was recorded on a professional grade hard-disk recorder. Thereafter, we ran the recorded clean guitar sound through a high-end all-tube amplifier connected to a speaker cabinet, in three different settings. In stu- dio terminology, this process is called “re-amplification” of the recorded signal. The re-amplified signal was recorded by a pro- fessional studio microphone, and stored in a hard-disk recorder. In this paper, we present a statistical analysis of certain aspects of the recorded guitar sound, for amplifier settings referred to as clean, crunch (half-distorted “rock’n roll” sound), and distortion (massively distorted “rock’n roll” sound). In this paper, we will approach the analysis and quantification of musical sounds from tube-amplified electric guitar, as a prob- lem in statistical signal processing. In this manner, we may utilize advanced and nonstandard theory to characterize emitted musical sound. As demonstrated, this may provide us with new and excit- ing ways to analyze musical time series, and we may in particular address issues of nonstationarity and nonlinearity in a systematic fashion. We believe this kind of analysis can be beneficial in the analysis of any kind of musical signals. 2. RANDOM PROCESSES Random processes are sequences of random variables. Music can be regarded as realizations of random processes. It is customary to assume some kind of statistical stationarity of random processes, as a simplifying approximation. Denote a real valued random processes by X(t), where t is a continuous or discrete time index. In general, the random process X(t) has the following spectral representation [4, 5] X(t)= Z e j2πft d e X(f ) (1) where d e X(f ) is the complex valued infinitesimal random Fourier generator (or the generalized Fourier transform) of the process X(t). If time is continous, the integration limits in Eq. (1) are ±∞, and if time is discrete, the limits are ±π/Δt, where Δt is the equidistant sampling interval. Since we assume X(t) ∈ R, the increment process has a useful Hermitian symmetry, d e X * (f )= d e X(-f ), where asterisk denotes complex conjugation. 2.1. Power spectrum The most common stationarity assumption, is that of Wide-Sense Stationarity (WSS). If the mean value function of the process is time invariant (or constant in time), μX(t)= E{X(t} = μ, and if the dual-time correlation function is independent on absolute or global time t, i.e., E{X(t +τ )X(t)}≡ R2(τ ), then the process is WSS. Here, E{·} denotes the statistical expectation operator. For stationary random processes the infinitesimal random Fourier gen- erator d e X(f ) are uncorrelated complex valued random variables with the following spectral correlation [4] E{d e X(f )d e X * (f )} = S2(f )df. (2) The well known Wiener-Khinchine theorem states that the power spectral density for WSS processes is related to the WSS autocor-