Digital filtering involves processing of sampled-data, or discrete-time, signals in accordance with a filtering algorithm. Digital filtering consists of taking what are typically equidistant discrete-time samples of a continuous-time function, or values of some discrete-time process, and performing operations such as discrete-time delay, multiplication by a constant and addition to obtain the desired result. A digital filter essentially comprises a series of elements between which signals are extracted, which signals are processed and are subsequently re-injected. A digital filter utilizes a computational process, carried out either through dedicated hardware or through the execution of a sequence of instructions by programmable logic, by way of which an input sequence of numbers representing discrete signal samples is converted into an output sequence of numbers, modified by the transfer function of the desired filter. Typical transfer functions refer to the frequency characteristics of the filter, examples of digital filter transfer functions include low-pass, high-pass, band-pass, etc. Digital filter computations typically include digital addition, digital multiplication of signal values by constants, and the insertion of delay stages. Digital filters are commonly used in processing digital data streams such as converted analog signals. Such digital filters often utilize a series of taps or weights assigned to each value in a series of data. These filters are often used in performing complex calculations for detecting certain conditions within the series of data. Digital filters are devices intended for converting a sampled signal, received as an input, into another sampled signal having predetermined frequency response characteristics. A sampled signal obviously includes a digital signal which is coded with a predetermined number n of bits on which the filter accuracy, or resolution, is dependent. Digital signal processors have a multiply-and-add function necessary for so-called filter processing for the realization of a digital filter. The digital signal processors adopt a stored program system employing microprograms and include an instruction ROM (read only memory) for storing microinstructions, and a data ROM as well as a data RAM (random access memory) for storing processing and other data. Digital filters can be used to process data from several sources or channels simultaneously. This is generally accomplished by applying samples from each of the sources to the filter in a predetermined sequence, such as by means of time division multiplexing of the samples. Digital filters have been broadly utilized in electronic appliances while A/D (analog to digital) converters on the basis of an oversampling scheme have become more prevalent. The analog to digital (A/D) conversion is a process for sampling analog signals in a discrete time and quantizing amplitude information to convert it into digital signals. The A/D converter on the basis of an oversampling scheme serves to convert analog input signals with a sampling rate which is by far higher than the highest frequency component of the analog input signals, and pass the digital output signals as converted through a digital low-pass filter which is located in a later stage in order to attenuate the noise level at high frequencies and lessen conversion noise.

Digital filters are often classified according to their impulse response. Such filters may be broadly divided into two lasses: infinite-impulse-response (IIR) filters and finite impulse response (FIR) filters. Finite impulse response (FIR) digital filters refer to the class of filters in which only a finite number of input samples affect the generation of a given output sample. An FIR digital filter is a time-invariant discrete linear system the output of which is determined by a weighted sum of a finite set of input samples, the weighting coefficients comprising the coefficients or weights of the filter defining its impulse response. A filter of this kind is usually called a non-recursive filter because its implementation does not require any feedback loop. Infinite impulse response (IIR) digital filters are a class of filters in which previous output samples are also used in generating a current output sample, and are thus typically realized in a recursive fashion, including feedback of output sample values. Because of the feedback of prior output values, each current output value of an IIR filter depends upon the value of an infinite series of input sample values. An infinite impulse response (IIR) filter generally achieves excellent amplitude characteristics by sacrificing phase characteristics. To obtain the same amplitude characteristics with a FIR filter as those of the IIR filter, many memories and coefficient multipliers are required. FIR filters have many advantages, including digital stability and phase linearity. An FIR filter is less complex to implement that an IIR filter having a comparable frequency characteristic. A finite impulse response (FIR) filter has a finite sequence of impulse responses and is generally non-recursive. Since a feedback loop is not needed in the non-recursive FIR filter, stability is guaranteed. Since it can satisfy the specification of linear phase characteristic specifications, the FIR filter is widely used in applications including waveform transmission. Wave digital filters (WDF) are a sub-class of digital filters which are considered to have a good dynamic range, low round-off noise, and inherent stability. Wave digital filters have very good stop band and pass band characteristics. These characteristics are particularly sensitive to small variations in coefficient values and offer a high degree of tolerance to non linear effects introduced by signal truncation and coefficient quantisation. WDFs are especially useful in digital processing when it is desired to minimize the number of bits used in the filter's coefficients. Among various digital filters, those in which the parameter for the predetermined operation is not fixed but varies over time such that intended output signals are obtained are called "adaptive digital filters".