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Recently, in a series of papers [Y. Rahimi, C. Wang, H. Dong, and Y. Lou, SIAM J. Sci. Comput., 41 (2019), pp. A3649-A3672; C. Wang, M. Tao, J. Nagy, and Y. Lou, SIAM J. Imaging Sci., 14 (2021), pp. 749-777; C. Wang, M. Yan, and Y. Lou, IEEE Trans. Signal Process., 68 (2020), pp. 2660-2669; P. Yin, E. Esser, and J. Xin, Commun. Inf. Syst., 14 (2014), pp. 87-109], the ratio of ℓ1 and ℓ2 norms was proposed as a sparsity inducing function for noiseless compressed sensing. In this paper, we further study properties of such model in the noiseless setting, and propose an algorithm for minimizing ℓ1/ℓ2 subject to noise in the measurements. Specifically, we show that the extended objective function (the sum of the objective and the indicator function of the constraint set) of the model in [Y. Rahimi, C. Wang, H. Dong, and Y. Lou, SIAM J. Sci. Comput., 41 (2019), pp. A3649-A3672] satisfies the Kurdyka--Lojasiewicz (KL) property with exponent 1/2; this allows us to establish linear convergence of the algorithm proposed in [C. Wang, M. Yan, and Y. Lou, IEEE Trans. Signal Process., 68 (2020), pp. 2660-2669] (see equation 11) under mild assumptions. We next extend the ℓ1/ℓ2 model to handle compressed sensing problems with noise. We establish the solution existence for some of these models under the spherical section property [S. A. Vavasis, Derivation of Compressive Sensing Theorems from the Spherical Section Property, University of Waterloo, 2009; Y. Zhang, J. Oper. Res. Soc. China, 1 (2013), pp. 79-105] and extend the algorithm in [C. Wang, M. Yan, and Y. Lou, IEEE Trans. Signal Process., 68 (2020), pp. 2660-2669] (see equation 11) by incorporating moving-balls-approximation techniques [A. Auslender, R. Shefi, and M. Teboulle, SIAM J. Optim., 20 (2010), pp. 3232-3259] for solving these problems. We prove the subsequential convergence of our algorithm under mild conditions and establish global convergence of the whole sequence generated by our algorithm by imposing additional KL and differentiability assumptions on a specially constructed potential function. Finally, we perform numerical experiments on robust compressed sensing and basis pursuit denoising with residual error measured by \ell 2 norm or Lorentzian norm via solving the corresponding ℓ1/ℓ2 models by our algorithm. Our numerical simulations show that our algorithm is able to recover the original sparse vectors with reasonable accuracy. [ABSTRACT FROM AUTHOR]