Some properties of a family of generalized time-limited window functions

For use in the design of finite impulse response (FIR) digital filters via the window function method the first integral of the window is required in order to relate transition width, filter order, and maximum passband and stopband error values. Again this relationship for the generalized family is found to be nearly linear if maximum error is measured in logarithmic units. Approximate empirical expressions are given for these relationships. Thus one can now design FIR filters with controlled error concentrated to any prescribed degree near to the band edges. Convenient computation methods for the generalized window functions are also described as well as the location of zeros and maxima and minima of their transforms.