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A new class of linearly tapered window, obtained by the convolution of a moving average filter and a running sum filter of unequal lengths, is introduced. The sidelobe rejection of the new window is between that of the rectangular and the Hann windows, and can be varied as a function of the lengths of the two convolving filters. We study the properties of the new window and provide empirical formulas for the sidelobe rejection and the mainlobe width. We propose rules for finite impulse response (FIR) filter design using the new window and show that a significant reduction in the filter length can be obtained as compared to that of the filter designed using the Hann window.