In a previous work, a solution to the fractional norm regularization (FNR) was discovered in a closed form and an inverse perturbation was adopted as a tool to overcome the ill condition of a matrix whose inverse is required by the fixed-point FNR. In this work, a novel computation technique, namely truncated singular value decomposition-fractional norm regularization (TSVD-FNR), is proposed for the recovery of a sparse signal in a compressive sensing problem. This estimation method can be used to reconstruct a signal measured from a large class of physical phenomena. Numerical simulation is accounted for both noiseless and noisy setups of the random signal and for a noiseless realistic signal framework derived from a beam vibration measurement. In comparing the proposed approach to previous methods, root-mean-square relative error (RMSRE) and elapsed time of computation are considered as performance metrics. For the noiseless case, it is shown herein that the TSVD-FNR algorithm significantly comes up with lower RMSRE than the previous algorithms for some regions of norm exponent, as long as the truncation ratio and the regularization parameter are suitably chosen. At the expense of more elapsed time of computation, the TSVD-FNR approach gives a perceptible lower RMSRE in the noisy case when the signal-to-noise ratio is high and the sparsity ratio is moderate. For recovering a realistic signal derived from the beam vibration measurement, the proposed TSVD-FNR method outperforms all other approaches in terms of normalized error.