In this work, we propose a novel approach called "Normalized Gain Function (NGF) method" to design low/medium power single stage ultra wide band microwave amplifiers based on linear S parameters of the active device. Normalized Gain Function T-NGF is defined as the ratio of T and vertical bar S-21 vertical bar(2), desired shape or frequency response of the gain function of the-amplifier to be designed and the shape of the transistor forward gain function, respectively. Synthesis of input/output matching networks (IMN/OMN) of the amplifier requires mathematically generated target gain functions to be tracked in two different nonlinear optimization processes. In this manner, NGF not only facilitates a mathematical base to share the amplifier gain function into such two distinct target gain functions, but also allows their precise computation in terms of T-NGF=T/vertical bar S-21 vertical bar(2) at the very beginning of the design. The particular ainplifier presented as the design example operates over 800-5200 MHz to target GSM, UNITS,. Wi-Fi and WiMAX applications. An SRFT (Simplified Real Frequency Technique) based design example supported by simulations in MWO (Micro Wave Office from AWR Corporation) is given using a 1400 mW pHEMT transistor, TGF2021-01 from TriQuint Semiconductor.