Headphones come in various different impedance levels, such as 8, 16 and 32 Ohms. The power supplied by the audio source can be at different levels due to a variety of factors such as being limited due to being battery powered. At lower power levels, lower impedance in the headphones can allow higher volume levels (sensitivity being equal).
From my basic electrical engineering courses, my instinct is that higher impedance headphones would also create a more level frequency response, thus being more true to the original signal. Is this true?
In relation to achieving optimal sound quality, is there any advantage to pairing high-impedance headphones with an amplifier if the audio source has a lower power level rather than simply using low-impedance headphones?