Inverse filters

Inverse filtering, or deconvolution as it is also known, is a very important branch of DSP. The principle can be understood by considering the following example. Imagine a signal that started life as a nice crisp waveform, but during transmission along a cable, it arrived at its destination in a somewhat smeared form, due to the limited frequency response of the cable. If this is known then the degraded signal can be filtered with the inverse of the cable’s frequency response, thereby reconstructing the original signal. Inverse filtering is widely applied, for example, to reconstruct old audio recordings, to perform frequency equalisation of loudspeaker systems and to enhance blurred astronomical images.

Signal Wizard can perform inverse filtering in real time, with the click of a few buttons. The frequency response of the degrading source is imported as a text file, and the inverse option selected. This generates an inverse filter which can now be downloaded to the hardware and executed on demand.

Case 1: Transmission line compensation in real time. In the leftmost plot are shown three sharp pulses; the plot in the middle shows the smearing of those three pulses having been transmitted through a cable with limited bandwidth, i.e. the cable acts like a low-pass filter. The plot on the right shows the pulses recovered in real time by Signal Wizard, using a filter with the inverse frequency response of the cable.

Case 2: Loudspeaker equalisation. Here, Signal Wizard is used to flatten the frequency response of a mid-range loudspeaker. It first probes the loudspeaker with a swept sine wave or white noise tone burst, and measures its frequency response via a microphone. The response is shown in the plot on the left. Note the uneven sensitivity. It then computes the inverse filter, which is now always included in the signal chain before the loudspeaker. The response is now considerably improved, as shown in the plot on the right.

Case 3: Audio reconstruction. In this example, a sound file has been degraded by re-sampling with a limited bandwidth and reducing to a single (mono) channel), simulating the effect of a low quality recording device. A high quality audio signal is recovered by feeding the signal through Signal Wizard running an inverse filter, and phasing the output for pseudo-stereo sound.