Demystifying digital signal processing for audio

By mic on 8/6/2012

We have been collecting some posts DSP and some DSP related wiki topics. Although we have a lot, we have been putting everything together a bit haphazardly. Anton Kamenov was nice enough to organize a lot of the DSP information on this website into a book: Digital Signal Procession for Audio Applications. The book is a very solid foundation for anyone who wants to know more about how DSP in audio comes about. It is well organized and derives a lot of the DSP operations that we take for granted in a simple and transparent manner. It also adds a lot of useful information – some that we have already started adding to this site and some that is still not up.

Here are some topics of interest in the book.

  1. Whether or not a practical complex signal consists of simple sine or cosine waves is not important. What is important is that we can approximate the signal for a finite period of time with its Fourier series expansion, which is a collection of simple waves.
  2. The Fourier transform is a natural extension of the Fourier series and is quite simple to use. This said, there are slight modifications to using the transform to derive finite impulse response filters or to gage the performance of these filters. For example, a good finite impulse response filter uses a generalized form of the inverse Fourier transform and not the inverse transform itself.
  3. The Fourier transforms (and inverse transforms) on real data produce redundant information.
  4. We can derive a simple good finite impulse response low pass filter in one of two ways. We can sum up the integer frequency cosine waves in the filter's pass band with the Dirichlet kernel. This is a "discrete time" form of the derivation, which works almost as well as the standard continuous derivation. The standard continuous derivation is to sample the inverse continuous Fourier transform of the desired frequency response.
  5. Finite impulse response filters exhibit the Gibbs phenomenon, which means that they have ripples close to the cutoff frequency. These ripples decrease in energy as the filter becomes more precise, but settle at a finite height. These ripples are the result of trying to approximate a discontinuous magnitude response with the continuous Fourier series and can easily be derived mathematically with the filter's transfer functions.
  6. Distortion introduces harmonics. Standard distortion techniques naturally introduce odd order harmonics, but there are ways to also introduce even order harmonics.
  7. The Z transform notation technically is used in digital signal processing only because it is shorter than the notation of the discrete-time Fourier transform. Other than that, we only rely on the discrete-time Fourier transform and we do not use the Z transform. The Laplace transform is only used because it produces nice infinite impulse response transfer functions.
  8. There are many errors that can be introduced by finite impulse response or infinite impulse response filters, although most errors have minimal impact on computations. Quantization errors are typically the hardest ones to get rid of, but one can use dithering. Dithering has many shapes and forms and the book provides some examples.
  9. Filters can be designed with optimization. The book concentrates on optimized IIR filters, as those require a more difficult optimization problem – we must be careful about filter stability and about the filters' phase responses when designing IIR filters.
  10. FIR filters and IIR filters can be combined. More interestingly, FIR filters can be undersampled when properly combined with IIR filters. The end result could be, for example, a short IIR filter with an extremely narrow transition band.
  11. Hilbert transforms are transforms that shift frequencies by a quarter of their cycle. They have practical applications as they allow us to measure the amplitude envelope of a signal. Finite impulse response Hilbert transforms are very easy to derive.
  12. Equiripple FIR filters can be designed with successive Fourier transform and inverse Fourier transform iterations.
  13. Natural reverbs are extremely complex sound phenomenon, which cannot be replicated artificially. We can, however, get very close with the Shroeder reverb.

The book's over 150 pages include many of the well-known DSP operations or phenomena: Butterworth filters, Chebychev type I and type II filters, Shroeder all pass and comb filters and the Shroeder reverb, feedforward and feedback comb filters, distortion, dithering, noise shaping, aliasing, the Gibbs phenomenon, the bilinear transformation. The book also includes a number of examples of some more esoteric DSP operations: the Shroeder-Moore low pass feedback comb filter, chorusing with delay sweeps, pitch shifting with the discrete Fourier transform, Hilbert transforms, impulse invariant filters, distortion with even order harmonics.

The book's index, table of contents, table of figures and example chapters are available for download on our Digital Signal Processing course page. The book can be purchased for $9.99 from that page as well, and from Amazon or Barnes & Noble (electronic for $9.99 or printed for about $40-$45).

authors: mic

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