Filters / Crossovers are not all Equal

This is under construction so bear with me until I get all my wording correct and try to condense this down into something that makes sense. This is highly mathematical and very difficult to explain in simple terms. I will try to define as much as possible. (for help see definitions below)

A crossover divides frequencies into sections that different speakers can handle. For example a small speaker like a 1" tweeter cannot reproduce low frequencies that a 12" subwoofer can handle.What is a crossover?

Why do I need one?

So you don't get distortion caused by the speaker failing to reproduce frequencies it wasn;t designed to handle. You get higher power handling, less distortion which sets you on a path for better sound.

Digital vs. Analog Crossovers

Analog crossoversare made of 4 electronic components -resistors, capacitors, inductors and opamps. The first three determine the value of the crossover and when used without opamps they arepassive. The 4 component is optional- Opamps are used inactivecrossovers or ones that arepoweredby an external power source. Analog crossovers have a 90 degree phase shift for every 6db/octave due to the properties of capacitors and inductors.

Digital Crossoversare peformed by mathematics. Depending on how they are designed, they can be modeled by the analog circuits (IIR) or more complex forms such as FIR filters. If they are modeled as analog crossovers in the digital domain, they still suffer from the 90 degree phase shift. It's easier to create filters with steeper cutoff's or higher order i.e. 60 or 72db/octave cutoff slope.

Digital IIR

Digital FIR

Examples

Definitions:

IIR

is a property of signal processing systems. Systems with that property are known as IIR systems or if we are dealing with electronic filter systems IIR filters. They have an impulse response function which is non-zero over an infinite length of time. This is in contrast to finite impulse response filters (FIR) which have fixed-duration impulse responses. The simplest analog IIR filter is an RC filter made up of a single resistor (R) feeding into a node shared with a single capacitor (C). This filter has an exponential impulse response characterized by an RC time constant.

FIR

is a type of a digital filter. It is 'finite' because its response to an impulse ultimately settles to zero. This is in contrast to infinite impulse response filters which have internal feedback and may continue to respond indefinitely.

Phase / Phase shift

Instantaneous phase

the current position in the cycle of something that changes cyclically

Phase shift

a constant difference/offset between two instantaneous phases, particularly when one is a standard reference

Discretization

concerns the process of transferring continuous models and equations into discrete counterparts. This process is usually carried out as a first step toward making them suitable for numerical evaluation and implementation on digital computers. In order to be processed on a digital computer another process named quantization is essential.

Quantization

is the process of approximating a continuous range of values (or a very large set of possible discrete values) by a relatively-small set of discrete symbols or integer values. More specifically, a signal can be multi-dimensional and quantization need not be applied to all dimensions. Discrete signals (a common mathematical model) need not be quantized, which can be a point of confusion.

Resistor / Resistance (R)

Resists the flow of current.

Inductor / Inductance (L)

is a measure of the amount of magnetic flux produced for a given electric current.

Capacitor / Capacitance (C)

is a measure of the amount of electric charge stored (or separated) for a given electric potential.

Operational Amplifier / Opamp

Frequency response

Impulse

Butterworth

Linkwitz Riley / LW4

Chebyshev

References

http://en.wikipedia.org/wiki/IIR

http://www.dspguru.com/info/faqs/iirfaq.htm

http://en.wikipedia.org/wiki/Finite_impulse_response

http://www.dspguru.com/info/faqs/firfaq.htm

http://en.wikipedia.org/wiki/Chebyshev_filter

http://en.wikipedia.org/wiki/Bessel_filter

http://en.wikipedia.org/wiki/Butterworth_filter

http://en.wikipedia.org/wiki/Comb_filter

http://en.wikipedia.org/wiki/Elliptic_filter

http://en.wikipedia.org/wiki/Quantiz..._processing%29

http://en.wikipedia.org/wiki/Discretization