Derivations and Relations of Various Cost Functions for Allpass PhaseEqualizing Filter Design This paper derives a new error function for the design of an allpass phaseequalizing filter (PEF). The new error function is nonlinear with respect to the unknown PEF coefficients, and minimizing the error function can obtain the unknown coefficients. Then, the relationship between the existing phaseerror function in the literature that is derived by using the Taylor series expansion and this new error function is revealed. Furthermore, the relationships among the new error function and two existing frequencyresponse error functions are derived. Therefore, this paper has two objectives. One is to develop a new error function for designing an allpass PEF, and the other is to reveal the relationships among the newly developed error function and various existing error functions, including a phaseerror function and two frequencyresponse error functions. Those error functions can be used as cost functions to be minimized in developing new optimization techniques for the design of an allpass PEF.

Flat PassBand Method with Two RC BandStop Filters for 4Stage Passive RC Polyphase Filter in LowIF Receiver Systems This paper proposes a flat passband for a 4stage passive RC polyphase filter in a bluetooth lowIF receiver system; there the bandwidth is 8MHz, the center IF frequency is 4MHz, and the required image rejection ratio is u003c30dB. Based on the superposition principle, the transfer function of this filter is derived. As the input signals are the wanted signals, there are two local maximum values which are calculated based on CauchySchwarz inequality theorem at 160kHz (1.24dB) and 40MHz (1.24dB). Therefore, two RC bandstop filters are used to improve the passband of these local maximum values (improvement of the ripple gain of passband from 2dB into 0.47dB). As a result, a 4stage passive RC polyphase filter for a lowIF receiver is designed, where an image rejection ratio is 36dB, and the passband gain is flat (the ripple gain is 0.47dB).