Introduction to Digital Signal Processing Using MATLAB with Application to Digital Communications pp 313-383 | Cite as

# Digital Filter Structures

## Abstract

What we have learnt so far is how to design either an IIR or FIR digital filter to satisfy a given set of specifications in the frequency domain. We have also seen examples based on MATLAB wherein filtering operations are carried out by specific functions. We really don’t know how these functions really work. If you are a S/W or H/W engineer and want to implement a digital filter in software or hardware, you should be able to describe the flow of signal from the input to the output. Thus, a digital filter structure describes the flow of signal as it propagates from the input to the output sample by sample. This filtering operation is described by a *signal flow graph*, which is a block diagram with blocks corresponding to the arithmetic operations of addition, multiplication, and unit delays. The blocks are connected by lines with arrows pointing in the direction of signal flow. In digital filter terminology, an adder has two inputs and one output, as shown in Fig. 8.1a. Similarly, a multiplier accepts an input signal and multiplies it by a coefficient *a* to produce an output, as shown in Fig. 8.1b. A unit delay block is a register, which can hold a sample from its input. The sample can be read from its output after one sample interval. Figure 8.1c illustrates a unit delay element. Note that the unit delay operation in the Z-domain is denoted by *z*^{−1}. Finally, Fig. 8.1d shows how a signal is tapped into. So, these are the basic building blocks of a digital filter structure. Let us look at a simple example.

## Supplementary material

## References

- 1.Agarwal RC, Burrus CS (1975) New recursive digital filter structures having very low sensitivity and roundoff noise. IEEE Trans Circ Syst, CAS-22(12): 921–927CrossRefGoogle Scholar
- 2.Burrus CS (1972) Block realization of digital filters. IEEE Trans Audio Electroacoust AU-20:230–235MathSciNetCrossRefGoogle Scholar
- 3.Buttner M (1977) Elimination of limit cycles in digital filters with very low increase in the quantization noise. IEEE Trans Circ Syst CAS-24:300–304CrossRefGoogle Scholar
- 4.Chan DSK, Rabiner LR (1973) Analysis of quantization errors in the direct form for finite impulse response digital filters. IEEE Trans Audio Electroacoust AU-21:354–366CrossRefGoogle Scholar
- 5.Chang T-L, White SA (1981) An error cancellation digital-filter structure and its distributed-arithmetic implementation. IEEE Trans Circ Syst CAS-28:339–342CrossRefGoogle Scholar
- 6.Classen TACM, Mecklenbrauker WFG, Peek JBH (1973) Some remarks on the classifications of limit cycles in digital filters. Philips Res Rep 28:297–305Google Scholar
- 7.Crochiere RE, Oppenheim AV (1975) Analysis of linear digital networks. Proc IEEE 62:581–595CrossRefGoogle Scholar
- 8.Dutta Roy SC (2007) A new canonic lattice realization of arbitrary FIR transfer functions. IETE J Res 53:13–18CrossRefGoogle Scholar
- 9.Dutta Roy SC (2008) A note on canonic lattice realization of arbitrary FIR transfer functions. IETE J Res 54:71–72CrossRefGoogle Scholar
- 10.Ebert PM, Mazo JE, Taylor MG (1969) Overflow oscillations in digital filters. Bell Syst Tech J 48:2999–3020CrossRefGoogle Scholar
- 11.Fettweis A (1971) Digital filter structures related to classical filter networks. Archiv fur Elektrotechnik und Ubertragungstechnik 25:79–81Google Scholar
- 12.Fettweis A (1975) On adapters for wave digital filters. IEEE Trans Acoust Speech Sig Process ASSP-23(6):516–525CrossRefGoogle Scholar
- 13.Gray AH Jr, Markel JD (1973) Digital lattice and ladder filter synthesis. IEEE Trans Audio Electroacoust AU-21:491–500CrossRefGoogle Scholar
- 14.Jackson LB (1969) An analysis of limit cycles due to multiplicative rounding in recursive digital filters. In: Proceedings, 7th Allerten conference on circuit and system theory, Monticello, IL, pp 69–78Google Scholar
- 15.Jackson LB (1970) On the interaction of roundoff noise and dynamic range in digital filters. Bell Syst Tech J 49:159–184MathSciNetCrossRefGoogle Scholar
- 16.Jackson LB (1970) Roundoff-noise analysis for fixed-point digital filters realized in cascade or parallel form. IEEE Trans Audio Electroacoust AU-18:107–122CrossRefGoogle Scholar
- 17.Jiang Z, Willson AN Jr (1997) Efficient digital filtering architectures using pipelining/interleaving. IEEE Trans Circuits Syst Part II 44:110–119CrossRefGoogle Scholar
- 18.Kan EPF, Aggarwal JK (1971) Error analysis in digital filters employing floating-point arithmetic. IEEE Trans Circ Theory CT-18:678–686CrossRefGoogle Scholar
- 19.Laroche L (1999) A modified lattice structure with pleasant scaling properties. IEEE Trans Sig Process 47:3423–3425CrossRefGoogle Scholar
- 20.Lawrence VB, Mina KV (1978) Control of limit cycle oscillations in second-order recursive digital filters using constrained random quantization. IEEE Trans Acoust Speech Sig Process ASSP-26:127–134CrossRefGoogle Scholar
- 21.Liu B, Kaneko T (1969) Error analysis of digital filters realized in floating-point arithmetic. Proc IEEE 57:1735–1747CrossRefGoogle Scholar
- 22.Long JJ, Trick TN (1973) An absolute bound on limit cycles due to roundoff errors in digital filters. IEEE Trans Audio Electroacoust AU-21:27–30CrossRefGoogle Scholar
- 23.Makhoul J (1978) A class of all-zero lattice digital filters: properties and applications. IEEE Trans Acoust Speech Sig Process 26:304–314CrossRefGoogle Scholar
- 24.Mills WL, Mullis CT, Roberts RA (1978) Digital filter realizations without overflow oscillations. IEEE Trans Acoust Speech Sig Process ASSP-26:334–338MathSciNetCrossRefGoogle Scholar
- 25.Mitra SK, Sherwood RJ (1973) Digital ladder networks. IEEE Trans Audio Electroacoust AU-21:30–36CrossRefGoogle Scholar
- 26.Mitra SK, Hirano K, Sakaguchi H (1974) A simple method of computing the input quantization and the multiplication round off errors in digital filters. IEEE Trans Acoust Speech Sig Process ASSP-22:326–329CrossRefGoogle Scholar
- 27.Mitra SK, Sherwood RJ (1974) Estimation of pole-zero displacements of a digital filter due to coefficient quantization. IEEE Trans Circ Syst CAS-21:116–124CrossRefGoogle Scholar
- 28.Mitra SK, Mondal K, Szczupak J (1977) An alternate parallel realization of digital transfer functions. Proc IEEE (Lett) 65:577–578CrossRefGoogle Scholar
- 29.Rabiner LR, Crochiere RE (1975) A novel implementation for narrow-band FIR digital filters. IEEE Trans Acoust Speech Sig Process 23(5):457–464CrossRefGoogle Scholar
- 30.Renner K, Gupta SC (1973) On the design of wave digital filters with low sensitivity properties. IEEE Trans Circ Theory CT-20:555–567CrossRefGoogle Scholar
- 31.Sandberg IW (1967) Floating-point-roundoff accumulation in digital filter realization. Bell Syst Tech J 46:1775–1791CrossRefGoogle Scholar
- 32.Swamy MNS, Thyagarajan KS (1975) A new type of wave digital filter. J Frankl Inst 300(1):41–58CrossRefGoogle Scholar
- 33.Szczupak J, Mitra SK (1975) Digital filter realization using successive multiplier – extraction approach. IEEE Trans Acoust Speech Sig Process ASSP-23:235–239CrossRefGoogle Scholar
- 34.Thyagarajan KS (1977) One and two-dimensional wave digital filters with low coefficient sensitivities. Ph.D. thesis, Concordia University, Montreal, Quebec, CanadaGoogle Scholar