Doppler spectrogram analysis of human gait via iterative adaptive approach L. Du, J. Li, P. Stoica, H. Ling and S.S. Ram Doppler spectrogram analysis of the human gait is a useful tool for discriminating various microDoppler tracks due to the movements of different body parts. A data-dependent algorithm, namely the short- time iterative adaptive approach, is used to obtain a more accurate spectrogram than the one provided by the conventional short-time Fourier transform-based approaches. The performance of the approaches is demonstrated and contrasted using both simulated and measured human gait data. Introduction: The Doppler spectrogram analysis of the human gait is useful in several applications including security and surveillance [1]. The returned radar signal from a moving human is nonstationary with time-varying frequency components. Doppler characteristics of the signal can be obtained by time–frequency analysis methods. Among these, the short-time Fourier transform (STFT) has been extensively used [2]. STFT assumes that the signal is stationary during a short time interval. The resulting spectrogram, i.e. the estimated signal power distri- bution in the time–frequency plane, contains the time-varying Doppler tracks due to the torso returns, as well as the weaker microDopplers due to the motion of arms and legs. However, it is well-known that STFT faces a difficult compromise: a shorter analysis window leads to better time resolution, but worse frequency resolution, and vice versa. This trade- off makes it hard to discriminate microDopplers in the STFT spectrogram, especially when the radar operating frequency is decreased to achieve wall penetration [1]. To overcome this limitation, the reassigned method [3–5] has been developed, and has been applied recently to microDoppler analy- sis of the human gait [1]. The method makes use of the phase information of STFT to obtain the instantaneous time t ins and instantaneous frequency f ins of the signal and then reassigns the value at each point in the spectro- gram to the corresponding position (t ins , f ins ). We propose an alternative method to improve the STFT spectrogram. Instead of using STFT, a novel short-time iterative adaptive approach (ST-IAA) is used to form the spectrogram. Owing to its adaptive (data-dependent) property, ST-IAA has much higher frequency resol- ution and lower sidelobes than STFT and thus ST-IAA provides much more accurate spectrograms. Moreover, the generalised information criterion (GIC) [6, 9] can be used in conjunction with ST-IAA to further improve the spectrogram quality. Problem formulation and ST-IAA: Let fy(n)g n¼1 N denote the data samples of the returned radar signal and let y(l ) ¼ [ y(P(l 2 1) þ 1) ... y(P(l 2 1) þ M )] T be the lth data frame with length M where P is a positive integer that denotes the number of non-overlapping samples between two consecutive frames (in the Example Section we use P ¼ M/10). Assuming that the frequency components of the signal are invariant over the interval of each data frame, a possible data model for y(l ) has the form: yðlÞ¼ Asðl Þþ eðl Þ; l ¼ 1; ... ; L ð1Þ where A W [a 1 a 2 ...a K ], e(l ) denotes a noise term and L ¼ b(N 2 M)/Pcþ 1 is the number of available frames. The column vectors in A can be expressed as a k ¼ [1 e j2pfk ...e j (M21)2pfk ] T , k ¼ 1, ... , K, where f k is the frequency value at the kth grid point, and K denotes the number of the uniform frequency grid points covering the frequency band [2f s /2, f s /2] ( f s is the sampling frequency). Finally s(l ) ¼ [s 1 (l ) ... s K (l )] T is the vector comprising the amplitudes of the possible frequency components in the lth data frame. Our goal is to estimate fs k (l )g and form an image of power distribution P k (l ) ¼ js k (l )j 2 , k ¼ 1, ... , K, l ¼ 1, ... , L, for each frequency grid point k and each time interval l. In what follows, we focus on the analysis of a generic data frame and the index l is omitted for notational simplicity. The ST-IAA algorithm estimates s k using the weighted least squares (WLS) criterion [6, 7]: ½y  a k s k   Q 1 k ½y  a k s k  ð2Þ where Q k is the interference (the frequency components in y at frequen- cies other than f k ) plus noise covariance matrix and is defined as follows: Q k ¼ R  P k a k a  k ð3Þ and R ¼ APA  with P denoting a K  K diagonal matrix, the diagonal of which contains the power at each frequency grid point fP k g k¼1 K . Minimising (2) with respect to s k yields: ^ s k ¼ða  k Q 1 k yÞ=ða  k Q 1 k a k Þ ð4Þ which would require the matrix inversion of Q k for every k. To reduce the computational complexity, we make use of the matrix inversion lemma and (see [6, 8]) to obtain an alternative expression for the WLS estimate: ^ s k ¼ða  k R 1 yÞ=ða  k R 1 a k Þ ð5Þ Since s ˆ k in (5) depends on R and therefore it depends on the very quantities fs k g that we want to estimate, (5) must be implemented in an iterative manner. This iterative process can be initialised via the esti- mate in (5) corresponding to R ¼ I, where I denotes the identity matrix. This initialisation is in fact an STFT with a rectangular window. The data-adaptive ST-IAA is expected to have a higher frequency resolution and lower sidelobes than the data-independent STFT, as suggested by the analysis of the stationary case in [6]. Usually, the number of the frequency grid points K is much larger than the actual number of frequency components. Therefore, the estimates obtained by ST-IAA for a given time interval generally yield some dominant peaks around the frequencies corresponding to the actual components. To discern the microDoppler characteristics more clearly, it may be desirable to show only the values of those peaks. To achieve this objective, we incorporate a reliable model- order selection tool, the GIC [6, 9] into ST-IAA. The result is what we call ST-IAA-GIC. To describe this algorithm briefly, let P denote the set of indices of the peaks ST-IAA of the spectrum and let I denote a subset of P. The GIC value for I is calculated as: GIC I ðhÞ¼ 2M ln ky  P j[I a j ^ s j k 2  ! þ rh lnð2M Þ ð6Þ where h is the size of I, k . k denotes the Euclidean norm, s ˆ j is obtained from ST-IAA for f j , and r is a parameter to control the penalty term, i.e. the second term on the right-hand side of (6), which is set to 0.1 in the following examples. The set I of significant peaks of the spectrum fP k ¼ js k j 2 g is determined by minimising GIC I (h) with respect to I , P. time, s frequency, Hz 0 0.5 1 1.5 2 2.5 -100 -80 -60 -40 -20 0 20 40 60 80 100 -120 -100 -80 -60 -40 -20 0 d 11 7 6 10 9 8 2 4 3 5 12 time, s frequency, Hz 0 0.5 1 1.5 2 2.5 -100 -80 -60 -40 -20 20 40 60 80 100 -120 -100 -80 -60 -40 -20 0 c 0 instantaneous time, s instantaneous frequency, Hz 0.5 1 1.5 2 2.5 -100 -80 -60 -40 -20 0 20 40 60 80 100 -120 -100 -80 -60 -40 -20 0 b 0 time, s frequency, Hz 0 0.5 1 1.5 2 2.5 -100 -80 -60 -40 -20 0 20 40 60 80 100 -120 -100 -80 -60 -40 -20 0 a Fig. 1 Spectrograms of simulated human gait data a STFT b Reassigned transform c ST-IAA d ST-IAA-GIC 1: Head 2: Torso 3: Right upper arm 4: Right lower arm 5: Left upper leg 6: Left lower leg 7: Left foot 8: Left upper arm 9: Left lower arm 10: Right upper leg 11: Right lower leg 11: Right foot Examples of human gait analysis: In the following examples, we apply ST-IAA and ST-IAA-GIC to the human gait analysis and compare the resulting spectrograms with those obtained via STFT and the reassigned ELECTRONICS LETTERS 29th January 2009 Vol. 45 No. 3