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140) CASE STUDY: LOW-ORDER FILTERS 35 where ωR is called the resonance frequency, similar to the case of analog filters. Note that in parallel RLC contours, the resonance appears at frequency ωR = 1 1 − LC 4C 2 R 2 Another important characteristic of digital resonators is amplification at the resonance frequency. 142) Another useful characteristic of filters is their bandwidth. 144) Note that efficiency of the filter is a continually increasing function when r → 1. 115) for narrowband digital resonators, the following simple √ formula can be used to evaluate the system bandwidth: ω ∼ = 2(1 − a2 ) = 2(1 − √ 1 − δ) ≈ δ.

In the general case, it is more convenient to use z-transform to determine y(n). 73) and, like the frequency response, fully describes a DLS. 74) ai z −i i=1 DLSs with N ≥ 1 are called recursive filters or filters with infinite impulse response (IIR). Value N , equal to the number of delay elements in the filter, is called the order of the filter. In contrast, if all ai = 0, then the filters are called transversal filters or filters with finite impulse response (FIR). 75) h(m) = 0 m>M Although the classification of the FIR and IIR filters considered here is broad, it is possible to find systems with N ≥ 1, but with a finite length of the impulse response.

7 In the general case, this corresponds to the exponentially dumped sinusoid, where r is the parameter responsible for the amplitude dumping. 9 and ϕ = π/2 is shown in Fig. 23. 2 Effects of Word Length Limitation When analysing DFs, we assume that neither the coefficients’ word length nor arithmetical operation processing are limited in bit number. In practice, the word length is always restricted, and investigation of the limits is an essential part of any DSP system design. Detailed analysis of this problem can be found in [1] and other sources.