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The tapered transmission line matching section is analyzed by treating it as a high pass filter. The ideal reflection coefficient characteristic for the taper that gives the smallest pass band tolerance for a given cutoff frequency and vice versa is derived from the expression for the reflection coefficient of the optimum designed n section quarter-wave transformer by taking the limit as n tends to infinity. By neglecting the square of the reflection coefficient in the differential equation for the reflection coefficient on the taper, a synthesis procedure is derived for obtaining an optimum taper. The procedure is similar to that used in designing an optimum line source distribution in antenna theory. Conditions are derived for the maximum allowable pass band reflection and change in impedance level for which the theory remains accurate. For a maimum reflection coefficient of 0.1 in the pass band the theory remains accurate for all frequencies above the cutoff value provided the change in impedance level does not exceed 7.5. This optimum taper is compared with the well-known exponential and Gaussian tapers and is found to be 13.9 per cent and 27 per cent shorter respectively for the same cutoff frequency and pass band tolerance.