The advent of the importance of Digital Signal Processing was heralded by the arrival of inexpensive, powerful microprocessors and the discovery of the fast fourier transform.  The ability to process a signal in real-time has led to the integration of DSP technology into countless areas of our lives.  Everything from   such devices as mobile phones, multimedia computers, video recorders, CD players, modems, and automobiles now utilize DSP’s in some fashion. 

 

An important operation in digital signal processing is that of filtering a signal.  Filtering is done for many reasons – filters can remove unwanted parts of the signal, such as random noise. They can extract useful parts of the signal, such as the components lying within a certain frequency range.  The general process used to filter analog signals is as follows:

1. Using an analog to digital converter, sample and digitize the analog signal.  The result will be stored as a sequence of bits. 
2. Apply the filter to the data.  This involves numerous multiplies (to find the product of the data sample and the filter coefficients) and adds (to find the sum of the current and previous weighted data samples). 
3. The filtered signal is then passed through a digital to analog converter to convert the signal back to its original analog form.

 

Thus, while analog filters work operate on voltages and currents, digital filters operate on sequences of bits. 

 

The “theory” behind a digital filter is as follows:

Suppose the signal that we wish to apply our digital filter to is originally in the form of a voltage waveform described by the function

V = x (t)

where t is time and can be any positive real number. 

This signal is sampled at time intervals h (the sampling interval). The sampled value at time t = kh (where k is a positive integer) is 

x_i = x (kh).

Thus the digital values returned by the ADC to be transferred to the DSP can be described by the sequence

x₀, x₁, x₂, x₃, ...

corresponding to the values of the signal waveform at times t = 0, h, 2h, 3h, ... (where t = 0 is the instant at which sampling begins).  Note that each x_n is represented by a sequence of bits. 

At time t = nh (where n is some positive integer), the values available to the processor, stored in memory, are

x₀, x₁, x₂, x₃, ... , x_n

Note that the sampled values x_n+1, x_n+2 etc. are not available as they occur in the future.  Thus, we cannot design anticipative digital filters, as they would require the knowledge of future sampled values of the signal.

The filter H is applied to each data sample.  In general, the value of the filtered signal at a particular instance depends on previous values of the signal.  A filter’s order describes how many previous inputs the output of the filter depends on.  For example, the output of a 2^nd order filter depends on the current signal sample and the two previous signal samples.  Thus, a filter’s order is an integer greater than or equal to zero. The digital output from the processor to the DAC consists of the sequence of values

y₀, y₁, y₂, y₃, ... , y_n