Recursive Digital Filters

1 Recursive Digital Filters A digital filter is the implementation of an algorithm that computes an output sequence of numbers from an input sequence of numbers. The algorithm may be implemented (or realized) using a general-purpose computer running software, a special-purpose computer such as a Digital Signal Processor (DSP) running software, or on special purpose hardware such as an Application Specific Integrated Circuit (ASIC) or Field Programmable Gate Array (FPGA). In this latter case, the digital circuit is designed specifically to accomplish the filtering task. This process is shown in Figure 0. Here, ( ) x n is an input sequence of numbers (usually, we read it as the n sample of the sequence x ), ( ) H z describes the recursive digital filter, and ( ) y n is the resulting output sequence of numbers. Whether the digital filter is implemented on a general-purpose computer, a special-purpose computer or with special purpose digital circuitry is usually a function of the required processing speeds (sampling rates) or the volume of units needed in production. The digital sequence of numbers ( ) x n is most typically generated from an analog waveform through the use of an analog-to-digital converter (A/D). The output sequence ( ) y n is most typically used to generate an analog waveform using a digital-to-analog converter (D/A). For instance, consider the recording and playback system for digital audio commonly called the compact disk. The audio is recorded using microphones, whose analog outputs are digitized using the A/D. These digital samples may be filtered in the processing and mixing of the recording. The resulting digital waveform is stored optically on the compact disk. The compact disk player reads the digital signal, filters these signals digitally, and then produces the analog output for the power amplifier and speakers using a D/A. Some signals are digital by nature, and thus do not require the conversions between the analog and digital formats. A classic example of such a signal is the price of a stock or commodity. Consider a digital speech signal that has been converted from an analog signal (a microphone output). The digital sequence representing the sentence “This is sound” is shown graphically in Figure 2. The amplitudes of the speech vary between 90 and 160, with the average value being approximately 128. This is the result of the speech being quantized to 8 bits, so that the amplitudes are limited to integers between 0 and 255 (which is 8 2 1 − ). The sampling frequency is 8 kHz, so that the samples are spaced 1 8000 seconds apart. The alternation pattern for