Time-frequency analysis of visual evoked potentials using chirplet transform J. Cui, W. Wong and S. Mann Chirplet time-frequency representation has been applied to charac- terise visual evoked potentials (VEPs) successfully. The approach presented here can represent both transient VEPs and steady-state VEPs clearly. Comparison to the method of short-time Fourier trans- form (STFT) is reported. Introduction: Detecting signals of visual evoked potentials (VEPs) elicited by repetitive stimuli is generally difficult, since the signal-to- noise ratio (SNR) of VEPs embedded in strong background noise and spontaneous EEG is rather low [1]. When the complete information of the signal to be detected is known, the optimal detector (in the Neyman-Pearson sense) is the likelihood ratio test which is usually implemented by a matched filter. Therefore, knowing the properties of the VEP signals related to a visual stimulus is important for designing detectors. Previous studies show that a steady-state VEP (ssVEP) is established if the repetition rate of visual stimuli is higher than some value and the shape of the resulting response becomes sinusoidal [1]. A transient process, however, precedes the formation of steady-state, characterised by abrupt changes of VEP amplitude within a short time interval. Under steady-state condition, the detection task can be reduced to finding a sinusoidal signal in noise by modelling the ssVEP signal as the summation of a fundamental frequency component and the higher harmonics, and ignoring the transient component. But because of the variability in the mental state of the subject (perhaps due to a lack of concentration, tiredness or accommodation), various factors can perturb the steady-state components. Moreover, from a physiological point of view, transient VEP appears to be more appropriate for rapid and reliable signal detection. Efforts have been made recently to characterise VEP signals over both its transient and steady-state portions [2]. Matching pursuit (MP) has been recently proposed as a nonlinear decomposition algorithm to decompose a signal into a very broad class of waveforms [3]. In MP, a sub-family of time-frequency atoms is chosen from the repertoire of the waveforms in such a way as to best match the local signal structure. In this Letter, we propose applying the method of MP algorithm using four-parameter chirplet atoms to do time-frequency analysis of VEPs. The purpose is to characterise both the transient and the steady-state of visual responses. Computational method: We propose a method whereby the VEP signal is decomposed over Gaussian chirplet atoms using the MP algorithm. A Gaussian chirplet atom is a four-parameter wave packet with a Gaussian envelope [4]: g b ðt Þ¼ 1 ffiffiffiffiffiffiffiffiffiffiffi ffiffiffi p p D t p exp  1 2 t  t c D t   2 ( þ j  2p½cðt  t c Þþ f c ðt  t c Þ  ð1Þ where j ¼ p (1), t c 2 R is the centre of the energy concentration in time, f c 2 R is the centre frequency, D t > 0 is the spread of the pulse, and chirprate c reflecting how quickly the chirp changes in time. The symbol b ¼ (t c , f c , D t , c) denotes the set of these four parameters. Our interest in using a Gaussian chirplet atom is mainly due to the fact that it is the function that has the highest joint time-frequency resolution and the only function whose Wigner distribution is non-negative. In practice, all four parameters should be discretised. The set of the parameter discretised atoms are called a dictionary. The first step (n ¼ 0) of the MP procedure is to choose the chirplet atom g b 0 from the dictionary so that the amplitude of the inner product (chirplet coefficient) jh f, g b 0 ij between this atom and signal f (t) is largest. Then the residual signal R 1 f, obtained after extracting the approximation of f in the direction of g b 0 from f, is decomposed in the similar way. Iterative procedures are applied to the subsequent residues: R 0 f ¼ f ; R nþ1 f ¼ R n f hR n f ; g b n i ( n 2 z ð2Þ In this way the signal f is decomposed into a sum of chirplet atoms that best match its residues: f ¼ P m n¼0 hR n f ; g b n i g b n þ R nþ1 f ð3Þ The amplitude of residue jR n f j decreases exponentially with each iterative step [3]. However, low amplitude residues may mainly be due to noise, and can be measured by correlation ratios [3]: lðR n f Þ¼ jhR n f ; g b n ij kR n f k ð4Þ That is, the larger the correlation ratios of the signal residues, the more coherent a residue and the less likely it is corrupted by noise. As a first-order approximation, the VEP signal is represented by a chirplet coefficient that results in the highest correlation ratio. From (4), to approximate R 0 f ¼ f with the highest correlation ratio is equivalent to selecting Gaussian chirplet atom g b 0 . The energy density of the approximated VEP signal in the time-frequency plane can be visualised by the Wigner distribution of the selected chirplet atom, i.e. g b 0 . Because their Wigner distributions do not include interference terms, they thus provide a clear picture in the time-frequency space. -50 50 40 30 20 10 0 -10 -20 -30 -40 VEP time, s 0 1 2 3 4 5 6 Fig. 1 Averaged VEPs for 6 s 0 20 15 10 5 frequency , Hz time, s time, s 3.0 3.2 3.4 3.6 3.8 4.0 4.2 4.4 4.6 4.8 5.0 b 0 60 40 20 frequency , Hz 0 1 2 3 4 5 6 a Fig. 2 Spectrogram of averaged VEP and windowed Fourier ridges of VEP signal between the third and fifth second a Spectrogram b Windowed Fourier ridges Results and discussion: The visual stimulus was presented as a sinusoidally oscillating single vertical-bar movement. The signal trace began with a 3 s interval without bar movement followed by another 3 s interval for the target signal. It is in this latter interval that the single horizontal bar undergoes a 3 Hz oscillatory motion. The third second of this 6 s epoch was called the onset of the stimulus. After amplification and filtering (lowpass filter at 40 Hz) the VEP data were sampled at 250 Hz and passed through an A=D converter. Fig. 1 shows the averaged VEP signal over 50 single sweeps. In Fig. 2, the spectrogram (STFT) of the averaged VEP is computed. The windowed Fourier ridges of the STFT spectrum are also shown in Fig. 2. In the spectrum, it is sufficient to distinguish the characters of ssVEP around 6 Hz where the second harmonic (2  3 Hz) is expected because of sufficient frequency resolution. However, the transient process of the VEP is blurred owing to poor time resolution. ELECTRONICS LETTERS 17th February 2005 Vol. 41 No. 4