Appendix 2 [Regarding approximations and, e.g., inequality (12)] FIR filter having sufficiently small errors in its passband We will frequently assess as "quite unlikely" that an FIR and stopband [e.g., to have practical implementation errors filter of a certain degree (order) can be realized for a par- in the passband and stopband that are sufficiently small, ticular example, or that a region of, e.g., the(x, y)-plane is compared with a desired min (d p, d s)], we know that W "practically unreachable" for a filter in a particular example. must be sufficiently large. By starting with the "approxi- Such references are intimately related to the notion of our mate" inequality (7), which expresses such a relationship digital filter being a "practical" FIR filter. In this regard in- and then, via (8) through (11), this requirement is embed- equality (7) shows a relationship between the wordlength ded into our fundamental (but necessarily "approximate") (W ) of the data words that are processed by a filter and a relation (12). Thus, FA + FF (expressing the total hardware desired passband and stopband quality of the filter. (Please required) must be large enough to accommodate a desired notice that it is pointed out in (7) that the relationship is (high enough) filter order, along with a desired high filter "approximate.") Fortunately, we are dealing with situations quality, expressed by min (d p, d s) . That is, there is a non- in which exact precision is not likely to be essential. We trivial (and likely not immediately obvious) interconnection are ultimately dealing with the approximate "cost" of imple- between the three entities: menting, in practice, a physical circuit. Other issues will of- * filter quality: ten come into the picture as well, and a "ball-park" estimate * size of filter (i.e., filter order, or degree): order min of a circuit's intrinsic implementation cost is all that one is * amount of physical hardware required: likely to require. This freedom allows us to arrive at "rather (A "threesome" not often found explicitly in FIR-filter simple" relationships between entities, relationships that work.) Our Fig. 3, for example, shows how these three fea- will surely suffice when, say, one must estimate the cost of tures interrelate, following reasonably simple, and useful, a proposed project. For example, in order to build a desired mathematical relationships, e.g. (12). and hence, the number of multiplier adders MA = 0. Therefore: Total HW complexity = FA + FF $ -^1 + a h ordermin log 2 ; min ^d p, d s h E. 7 (11) From (11) we obtain: ordermin # FA + FF min ^d p, d s h E -^1 + a h log 2 ; 7 (12a) ordermin # HW budget . min ^d p, d s h E -^1 + a h log 2 ; 7 (12b) The outcome in (11) and (12) predicts that, given a fixed hardware budget, in terms of the total number of full adders and flip-flops, a practical realization of an FIR filter with a Remez order larger than the upper bound defined in (12) is highly improbable (see Appendix 2)-where the parameter a is defined in (2). The practical hardware complexity bound in (12) is visually illustrated in Fig. 3 both in linear and log scales. The z-axis represents 14 IEEE cIrcuIts ANd systEMs MAgAzINE FA + FF the highest Remez order of the practically realizable FIR filter, as a function of the hardware budget (x-axis) and filter spec (y-axis) . These results are further illustrated using contour plots in Fig. 4. The predictive capabilities of the proposed upper bound, as defined in (11) and (12), will be discussed in the following sections. While the choice of parameter a = 1 is appropriate for the majority of FIR filter design cases, we offer an extension of (2) for a more conservative definition of parameter a: a . C#' or, alternatively: min (d p, d s) 1 - 1/M 1 for Mth band FIR filters otherwise (13) where scaling factor C is a function of ~ p (0 < ~ p < 1) to take into account the fact that the average coefficient complexity is more sensitive to the filter's passband zeros than to its stopband zeros [46], [47]. The extension in (13) provides a slightly more conservative prediction for extremely narrowband filters (i.e., ~ p % 1) . To further extend the result in (11) so that the minimum required total HW complexity is defined directly as a function of the filter specifications, we can use Kaiser's approximation (14) for the FIR filter's order [35], [41]. Alternatively, filter-order prediction theories, including Bellanger's formula [35], the Mintzer-Liu estimation [35], fIrst quArtEr 2018

IEEE Circuits and Systems Magazine - Q1 2018 - Cover1

IEEE Circuits and Systems Magazine - Q1 2018 - Cover2

IEEE Circuits and Systems Magazine - Q1 2018 - Contents

IEEE Circuits and Systems Magazine - Q1 2018 - 2

IEEE Circuits and Systems Magazine - Q1 2018 - 3

IEEE Circuits and Systems Magazine - Q1 2018 - 4

IEEE Circuits and Systems Magazine - Q1 2018 - 5

IEEE Circuits and Systems Magazine - Q1 2018 - 6

IEEE Circuits and Systems Magazine - Q1 2018 - 7

IEEE Circuits and Systems Magazine - Q1 2018 - 8

IEEE Circuits and Systems Magazine - Q1 2018 - 9

IEEE Circuits and Systems Magazine - Q1 2018 - 10

IEEE Circuits and Systems Magazine - Q1 2018 - 11

IEEE Circuits and Systems Magazine - Q1 2018 - 12

IEEE Circuits and Systems Magazine - Q1 2018 - 13

IEEE Circuits and Systems Magazine - Q1 2018 - 14

IEEE Circuits and Systems Magazine - Q1 2018 - 15

IEEE Circuits and Systems Magazine - Q1 2018 - 16

IEEE Circuits and Systems Magazine - Q1 2018 - 17

IEEE Circuits and Systems Magazine - Q1 2018 - 18

IEEE Circuits and Systems Magazine - Q1 2018 - 19

IEEE Circuits and Systems Magazine - Q1 2018 - 20

IEEE Circuits and Systems Magazine - Q1 2018 - 21

IEEE Circuits and Systems Magazine - Q1 2018 - 22

IEEE Circuits and Systems Magazine - Q1 2018 - 23

IEEE Circuits and Systems Magazine - Q1 2018 - 24

IEEE Circuits and Systems Magazine - Q1 2018 - 25

IEEE Circuits and Systems Magazine - Q1 2018 - 26

IEEE Circuits and Systems Magazine - Q1 2018 - 27

IEEE Circuits and Systems Magazine - Q1 2018 - 28

IEEE Circuits and Systems Magazine - Q1 2018 - 29

IEEE Circuits and Systems Magazine - Q1 2018 - 30

IEEE Circuits and Systems Magazine - Q1 2018 - 31

IEEE Circuits and Systems Magazine - Q1 2018 - 32

IEEE Circuits and Systems Magazine - Q1 2018 - 33

IEEE Circuits and Systems Magazine - Q1 2018 - 34

IEEE Circuits and Systems Magazine - Q1 2018 - 35

IEEE Circuits and Systems Magazine - Q1 2018 - 36

IEEE Circuits and Systems Magazine - Q1 2018 - 37

IEEE Circuits and Systems Magazine - Q1 2018 - 38

IEEE Circuits and Systems Magazine - Q1 2018 - 39

IEEE Circuits and Systems Magazine - Q1 2018 - 40

IEEE Circuits and Systems Magazine - Q1 2018 - 41

IEEE Circuits and Systems Magazine - Q1 2018 - 42

IEEE Circuits and Systems Magazine - Q1 2018 - 43

IEEE Circuits and Systems Magazine - Q1 2018 - 44

IEEE Circuits and Systems Magazine - Q1 2018 - Cover3

IEEE Circuits and Systems Magazine - Q1 2018 - Cover4

https://www.nxtbook.com/nxtbooks/ieee/circuitsandsystems_2023Q3

https://www.nxtbook.com/nxtbooks/ieee/circuitsandsystems_2023Q2

https://www.nxtbook.com/nxtbooks/ieee/circuitsandsystems_2023Q1

https://www.nxtbook.com/nxtbooks/ieee/circuitsandsystems_2022Q4

https://www.nxtbook.com/nxtbooks/ieee/circuitsandsystems_2022Q3

https://www.nxtbook.com/nxtbooks/ieee/circuitsandsystems_2022Q2

https://www.nxtbook.com/nxtbooks/ieee/circuitsandsystems_2022Q1

https://www.nxtbook.com/nxtbooks/ieee/circuitsandsystems_2021Q4

https://www.nxtbook.com/nxtbooks/ieee/circuitsandsystems_2021q3

https://www.nxtbook.com/nxtbooks/ieee/circuitsandsystems_2021q2

https://www.nxtbook.com/nxtbooks/ieee/circuitsandsystems_2021q1

https://www.nxtbook.com/nxtbooks/ieee/circuitsandsystems_2020q4

https://www.nxtbook.com/nxtbooks/ieee/circuitsandsystems_2020q3

https://www.nxtbook.com/nxtbooks/ieee/circuitsandsystems_2020q2

https://www.nxtbook.com/nxtbooks/ieee/circuitsandsystems_2020q1

https://www.nxtbook.com/nxtbooks/ieee/circuitsandsystems_2019q4

https://www.nxtbook.com/nxtbooks/ieee/circuitsandsystems_2019q3

https://www.nxtbook.com/nxtbooks/ieee/circuitsandsystems_2019q2

https://www.nxtbook.com/nxtbooks/ieee/circuitsandsystems_2019q1

https://www.nxtbook.com/nxtbooks/ieee/circuitsandsystems_2018q4

https://www.nxtbook.com/nxtbooks/ieee/circuitsandsystems_2018q3

https://www.nxtbook.com/nxtbooks/ieee/circuitsandsystems_2018q2

https://www.nxtbook.com/nxtbooks/ieee/circuitsandsystems_2018q1

https://www.nxtbookmedia.com