What is the voltage and current rating of the supply? Is there any other load on the supply? Is the supply a regulated or unregulated supply?

Most of these supplies require a minimum load before the output voltage is within a specified range. Other than the schmitt-triger was there any other load on the supply? If not you are probably not loading the supply to the specified minimum current.

In most applications these supplies are followed with a linear regulator to provide a stable voltage. You still need a minimum load and some filtering capacitors. 4000uF is excessive and possibly dangerous. Typical capacitors are in the 1-100uF range. The size and type of capacitor is dependent on the linear regulator. Follow the manufacturers ecommendation exactly. Some devices have very specific capacitance and ESR (effective series resistance) requirements.

(* jcl *)

http://www.wiblocks.com

Sparkfun has a good tutorial on power supplies that covers these types of issues.

In my understanding, the more load you draw off a power supply, the less effective a particular-sized capacitor will be at stabilising the voltage. So, if you had a low load, the current capacitor might be fine, but if the "radio" had a decent amplifier and speaker incorporated the load might be enough to reduce the capacitor's effectiveness and cause noise to make its way into the circuit. The solution in that case is to increase the capacitor size.

If the DC output from the adapter is higher than what is needed in the circuit, another option that will help stabilise the output is to regulate the voltage down as required, after the capacitor, using a standard voltage regulator.

Make sure you use a power supply which is properly regulated for your application. (jluciani's comments are good ones) Ripple voltage should be one of the specs.

If it's not good enough for your purposes, get a different one -- you can't really fix a power supply in a way that's guaranteed to work in all cases.