The thesis presents algorithms based on a linear algebraic solution for the identification of the parameters of the FIR system using only higher order statistics when only the output of the system corrupted by additive Gaussian noise is observed. All the traditional parametric methods of estimating the parameters of the system have been based on the 2nd order statistics of the output of the system. These methods suffer from the deficiency that they do not preserve the phase response of the system and hence cannot identify non-minimum phase systems. To circumvent this problem, higher order statistics which preserve the phase characteristics of a process and hence are able to identify a non-minimum phase system and also are insensitive to additive Gaussian noise have been used in recent years. Existing algorithms for the identification of the FIR parameters based on the higher order cumulants use the autocorrelation sequence as well and give erroneous results in the presence of additive colored Gaussian noise. This problem can be overcome by obtaining algorithms which do not utilize the 2nd order statistics. An existing relationship between the 2nd order and any Ith order cumulants is generalized to a relationship between any two arbitrary k, Ith order cumulants. This new relationship is used to obtain new algorithms for FIR system identification which use only cumulants of order > 2 and with no other restriction than the Gaussian nature of the additive noise sequence. Simulation studies are presented to demonstrate the failure of the existing algorithms when the imposed constraints on the 2nd order statistics of the additive noise are violated while the proposed algorithms perform very well and give consistent results. Recently, a new algebraic approach for parameter estimation method denoted the Linear Combination of Slices (LCS) method was proposed and was based on expressing the FIR parameters as a linear combination of the cumulant slices. The rank deficient cumulant matrix S formed in the LCS method can be expressed as a product of matrices which have a certain structure. The orthogonality property of the subspace orthogonal to S and the range space of S has been exploited to obtain a new class of algorithms for the estimation of the parameters of a FIR system. Numerical simulation studies have been carried out to demonstrate the good behaviour of the proposed algorithms. Analytical expressions for the covariance of the estimates of the FIR parameters of the different algorithms presented in the thesis have been obtained and numerical comparison has been done for specific cases. Numerical examples to demonstrate the application of the proposed algorithms for channel equalization in data communication and as an initial solution to the cumulant matching nonlinear optimization methods have been presented.