An important issue and source of error with respect to digital filtering (and digital systems in general) is the finite word lengths used to represent data.  The finite number of bits available to represent data leads to an inherent lack of precision.  There are several places in which these errors can effect the processing of digital signals. 

1. If we are performing digital filtering of analog signals, the analog signal must pass through an analog to digital converter (ADC), where the signal is quantized.  This quantization will cause some accuracy to be lost. 
2. The coefficients used to specify the behavior of the filter are usually real numbers.  Again, the finite number of available bits limits our ability to enter these coefficients exactly.  Instead, the coefficients will be rounded to the nearest representable number, causing the filter to behave slightly differently from its specification.
3. When calculating the output signal, a signal data point is multiplied by the filter coefficient.  The result of this multiplication will sometimes be too long to fit into the original word size.  Thus, it must be truncated and rounded.  This product round off will lead to some additional imprecision. 

 

The DSP functions that we use require that their input data be represented in Q0.15 format, though it is more commonly referred to as Q15.  In a Qm.n format, there are m bits used to represent the two's complement integer portion of the number, and n bits used to represent the two's complement fractional portion.  Also, m+n+1 bits are needed to store a general Qm.n number.  The extra bit is needed to store the sign of the number in the most-significant bit position.  The representable integer range is specified by  and the finest fractional resolution is .  Thus, each Q15 data word is 16 bits long.  The leftmost bit is the sign bit and the rightmost 15 bits are used to represent the two’s complement fractional part of the number.  The allowable range of numbers is (-1, 1) and the smallest fractional resolution is 2^-15 or approximately 3.05 * 10^-5.