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In this paper, we analyze Raman amplifiers (RAs) with time division multiplexed (TDM) pumps via an analytical approach. First of all, a detailed analytical solution for the forward and backward propagating waves in the TDM-pumped RAs is presented. Then, the gain of the TDM-pumped RAs is treated analytically. After revealing the nature of resonant gain enhancement, a relatively simple formula of the optimal input pump power configuration is derived via the least squares method. Afterwards, enhanced double Rayleigh backscattering noise and amplified spontaneous emission noise are analyzed and explicit analytical formulas derived. Thereafter, a simple formula to evaluate the impact of the pump modulation is proposed for the design, analysis, and optimization. The analysis of the TDM-pumped RAs highlights the optimization of the parameters of the TDM pumps, such as the pump wavelengths, pump power, and pump modulation frequency as well as the pumping order.