Filtering power supply noise

I am getting 7 volts from a power supply source which I need to step down to 3.3 V for powering up my 2.4 GHz zigbee MCU (TI CC2530 + CC2592 amplifier).

I am using a linear regulator to step it down. My schematic is as shown below:

Randomly I am seeing MCU resets and brief communication failures. I checked the power line waveforms and they look like these (with ac coupling):

Input (7 V line):

FFT looks like this:

Output (3.3 V line)

FFT looks like this:

I can do nothing about the input. What should I do to ensure a better power line for the MCU in order to get rid of resets and communication failures.

Note - same circuit works fine if I power it up from my bench power supply.

EDIT (solution posted below):

I followed @analog's solution but with different values (because I didn't have the values that he mentioned):

Setup:

Waveforms with 180 ohm load on the 7 V line (ac coupling):

Waveforms with 33 ohm load on the 7 V line (First one is DC coupling and rest are ac coupling):

EDIT: Adding more waveforms.

Both probes connected at Vout. Tested at no load and 33 ohm load conditions. Yellow line is DC coupling. Blue line is ac coupling. No load voltage is roughly 7.5 volts. At 33 ohm load, the voltage drops down to roughly 5.7 volts.:

Answer

Like this

^{simulate this circuit – Schematic created using CircuitLab}

If your current demand will not allow the 10 ohm resistor, then use 1uH inductor in parallel with 100 ohms Rdampen.