We develop characteristics-based filter design methods for a class of IIR bandpass filters, which we refer to as Generalized-Exponent Filters (GEFs) and that are represented as second-order filters raised to non-unitary exponents. GEFs have a peak, are effectively linear phase, and are useful for seismic signal phase-picking, cochlear implants, and equalizers. The native frequency-domain specifications for GEFs are not on given frequency responses but rather on filter characteristics such as peak frequency, bandwidth, and group delay. Our characteristics-based method for filter design accommodates direct specification of a trio of frequency-domain characteristics from amongst the peak frequency, convexity, ndB quality factors, equivalent rectangular bandwidth, maximum group delay, and phase accumulation. We achieve this by deriving filter parameterizations with sets of filter characteristics which involves deriving closed-form analytic expressions mapping sets of filter characteristics to the original filter constants by making sharp-filter approximations. This results in parameterizations for GEFs including ones with simultaneous specification of magnitude-based and phase-based characteristics (e.g. bandwidths and group delays). This in turn enables designing sharply tuned filters without significant group delay, and simultaneous control over frequency selectivity and synchronization which is important in designing filterbanks. Our filter design methods with direct control over characteristics may also be utilized beyond static filter design for higher-order variable bandpass filter design and may be useful for characteristics-based adaptive filtering. Our methods are inherently stable, highly accurate in meeting strict specifications on desired characteristics, simple, and computationally efficient. The methods extend to the design of related bandpass and multiband filters.