Reconstructive Filtering

For starters, I made an op amp in an inverting configuration with a gain of 2. The op amp had rails of +/- 5 V, with an input of a 1 KHz sine wave, at 6 Volts peak to peak.

Source Signal

Taking a FFT of the input purely for comparison will give a single harmonic with an amplitude of 3V, at 1 KHz as expected.

FFT of Input Signal

Schematic of Clipping Circuit

Based on the gain and the rails, along with the input signal, it is clear that there will be a good amount of clipping on the signal.

The above circuit gives the following Bode Plot:

Bode Plot of Clipping Circuit

After running the simulation the following output signal was observed:

Transient of Clipping Circuit

This output signal slightly resembles a square wave, which has a Trigonometric Fourier Transform of:

Fourier Transform of a Square Wave

This can be easily observed from the nature of the function as there is no DC biasing (in simulation), eliminating any DC component and the function is odd, thus eliminating the cosine component. This produced the following FFT.

FFT of Clipped Signal

From Fourier analysis, the clipping has added additional harmonics outside of the fundamental frequency of 1KHz.

This was experimentally observed using an LM-324 Op-Amp.

Experimentally Observed Clipped Transient

Due to the skew in the observed signal, the gradual changes in the signal prevent the zeroing out of the even harmonics, thus producing the below Fourier Representation.

Experimentally Observed Clipped FFT

So what can be done about the clipping? Well, we need to attenuate the harmonics by a lot to get back to a clean signal, implying the need for a sharp filter (maybe a third order Low Pass Filter). Below is the schematic for a simple, active third order LPF which was connected to the clipping amplifier. Since the active filter will have +/- 15 V rails, it is safe to assume that there will be no clipping, thus I used an ideal voltage controlled voltage source (VCVS) to act as an Operational Amplifier.

Clipped Circuit combined with a Third Order LPF

The filter has a break frequency of roughly 1KHz, as shown in the Frequency Analysis shown below. The Bode Plot also included the clipped signal, which is shown in blue.

Bode Plot of Clipping Stage and LPF

The Transient of the new circuit gives promising results, generating an output wave similar to that of a sine wave. The red line is the output of the third order LPF and the blue is the source signal applied to the circuit. Due to the nature of the filter, a 90 degree phase shift was observed.

Reconstructed Transient in Simulation

To see how close the output signal was to a perfect sine wave, a Fourier Analysis was run against the resulting signal producing:

FFT of Reconstructed Transient

The resulting signal is extremely close with the only noticeable additional harmonic occurring at 3KHz.

The reconstruction filter was built and below is the resulting transients and frequency response.

Transient of Experimentally Observed Reconstructed Signal

FFT of Experimentally Observed Reconstructed Signal