Effect of Random Sampling on Noisy Nonsparse Signals in Time-Frequency Analysis Isidora Stankovi´ c GIPSA Lab/Faculty of Electrical Engineering University Grenoble Alpes/University of Montenegro Grenoble, France/Podgorica, Montenegro isidoras@ac.me Miloˇ s Brajovi´ c, Miloˇ s Dakovi´ c Faculty of Electrical Engineering University of Montenegro Podgorica, Montenegro {milosb,milos}@ac.me Cornel Ioana GIPSA Lab University of Grenoble Alpes Grenoble, France cornel.ioana@gipsa-lab.grenoble-inp.fr Abstract—The paper examines the exact error of randomly sampled reconstructed nonsparse signals having a sparsity con- straint. When signal is randomly sampled, it looses the property of sparsity. It is considered that the signal is reconstructed as sparse in the joint time-frequency domain. Under this as- sumption, the signal can be reconstructed by a reduced set of measurements. It is shown that the error can be calculated from the unavailable samples and assumed sparsity. Unavailable samples degrade the sparsity constraint. The error is examined on nonstationary signals, with the short-time Fourier transform acting as a representative domain of signal sparsity. The pre- sented theory is verified on numerical examples. Index Terms—compressive sensing, nonsparse signals, random sampling, time-frequency analysis I. I NTRODUCTION Nonstationary signals are dense in both time and frequency, when considered separately. They can be localized in the time- frequency domains. However, they could be located within much smaller regions in the joint domain using appropriate representations [1]–[6], with the short-time Fourier transform (STFT) being the basic transformation. The signals are sparse in the time-frequency domain if the number of nonzero coef- ficients in this domain is much smaller than the total number of coefficients. According to the compressive sensing (CS) theory, sparse signals can be reconstructed using less samples/measurements than required by the sampling theorem [7]–[12]. Reducing the number of measurements will introduce noise in the analysis of the signals. The properties of the noise from [13], [14] will be used to define reconstruction properties in the case of randomly sampled STFT. If a nonsparse signal is reconstructed with a reduced set of available samples, then the noise due to the missing samples of nonreconstructed coefficients will be considered as an additive input noise in the reconstructed signal. Because of its nonstationary nature, signals in time- frequency domain are usually approximately sparse or non- sparse. In the CS literature, only the general bounds for the reconstruction error for nonsparse signals (reconstructed with the sparsity assumption) are derived [9], [15]–[17]. In this paper, we present an exact relation for the expected squared error, reconstructed from a reduced set of signal samples, under the sparsity constraint. The error depends on the number of available samples and the assumed sparsity. In order to be more compatible with the practical problems, we will consider that the signals are randomly sampled, i.e. not on the grid. Also, signals with additive noise will be considered. Since the signal is not on the grid, it looses the property of sparsity in the transformation domain. The properties of uniform sampling without noise are examined in [18]. The effects of random sampling and noise are illustrated and checked on examples. The paper is organized as follows. The theory of random sampling in time-frequency analysis using the compressive sensing framework will be explained in Section II. The influ- ence of nonsparsity in randomly sampled signal will be shown in Section III. Examples will be given in Section IV and the conclusions are presented in Section V. II. THEORETICAL BACKGROUND A. Random sampling Consider a general form of a multicomponent signal x(t)= C  l=1 x l (t), (1) with C non-stationary components x l (t), l =1, 2,...,C . The signal is of a time-varying nature. Although not sparse in the Fourier transform (FT) domain, it may be sparse in the joint time-frequency domain. In this paper, we assume that the signal is sparse in the STFT domain, which is defined as S N (t, Ω) =  ∞ -∞ x(t + τ )w(τ )e -jΩτ dτ, (2) where w(τ ) is the window function with duration T , centered at point t. The periodic extension of the product x(t + τ )w(τ ), for a given t can be expanded in Fourier series as follows x(t + τ )w(τ )= 1 N N-1  k=0 X t (k)e j2πk(τ -T/2)/T , (3) with series coefficients X t (k) being equal to the discrete FT (DFT) coefficients if x(n+m)w(m) is used to denote x(nΔt+ 2018 26th European Signal Processing Conference (EUSIPCO) ISBN 978-90-827970-1-5 © EURASIP 2018 485