Complex Signal Processing Is Not — Complex Ken Martin [email protected] Dept

Complex Signal Processing is Not — Complex Ken Martin [email protected] Dept. of Elect. and Comp. Engr., Univ. of Toronto, Toronto, ON Canada M5S 3G4 Wireless systems often make use of the quadrature Abstract: x ()t y ()t relationship between pairs of signals to effectively cancel r r out-of-band and interfering in-band signal components. The cos()ωit understanding of these systems is often simplified by consider- y ()t ing both the signals and system transfer-functions as ‘complex’ q quantities. The complex approach is especially useful in sin()ωit highly-integrated multi-standard receivers where the use of (a) narrow-band fixed-coefficient filters at the RF and high IF frequencies must be minimized. A tutorial review of complex sig- Xt()= xr()t + j× 0 Yt() nal processing for wireless applications emphasizing a jω t graphical and pictorial description rather than an equa- e i tion-based approach is first presented. Next, a number of clas- (b) sical modulation architectures are described using this Fig. 1. A signal-flow-graph (SFG) of an image-reject mixer formulation. Finally, more recent developments such as com- using (a) a real SFG and (b) a complex SFG. plex filters, image-reject mixers, low-IF receivers, and then taking the Fourier Transform we have over-sampling A/D converters are discussed. 1. Introduction Y()ω = X()ωω– i (2) A complex signal consists of a pair of real signals at an and we see that an ideal image-reject mixer results in a simple instance in time. If one denotes the complex signal, Xt(), as frequency-shift of the input signal with no images occurring, a Xt()= xr()t + jxq()t , where j1= – , then a Hilbert Space conclusion that is obvious from Fig. 1. (b), but is certainly not can be defined using appropriate definitions for addition, mul- obvious from Fig. 1. (a). tiplication, and an inner-product and norm. In an actual physi- The use of complex signal processing for wireless applica- cal system, the signals are both real (but are called the real and tions has blossomed recently. This is especially true for imaginary parts) and are found in two distinct signal paths. The high-bit-rate standards, such as local-area-networks (LAN’s), multiplier ‘j ’ is used to help define operations between differ- and for multi-standard transceivers. Evidence of this prolifera- ent complex signals, and is to some degree imaginary, i.e., it tion is that in 2001, 2002, and 2003, the majority of the papers doesn’t actually exist in real systems. Often the dependence of in IEEE ISSCC sessions on wireless transceivers for LAN signals on time is not shown explicitly. applications employed complex signal processing. The use of complex signal processing to describe wireless Much, but not all, of this paper deals with a tutorial review systems is increasingly important and ubiquitous for the fol- of complex signal processing, especially as applicable to wire- lowing reasons: it often allows for image-reject architectures to less systems. A great deal of the tutorial material on complex be described more compactly and simply, it leads to a graphical analog signal processing is directly based on the pioneering or signal-flow-graph (SFG) description of signal-processing research of W. Martin Snelgrove as described in [1][6][7], and systems that gives insight, and it often leads to the develop- in some theses of his graduate students in [9] and [11]. Unfor- ment of new systems where the use of high-frequency, highly tunately, no journal publications of this early work were pub- selective image-reject filters is minimized leading to more lished1. Some of the early published work on complex digital highly integrated transceivers using less power and requiring filters is described in [2] and [5]. It is speculated that the devel- less physical space. opment of complex filters was partially due to the use of For- As an example, consider the popular ‘image-reject’ mixer tran to simulate and design filters and the fact that Fortran having a real input as shown in Fig. 1. In the complex SFG, could do complex math — an example of a solution finding a two real multiplications have been replaced by a single com- problem. plex multiplication. Furthermore, since in the time domain we This paper starts with a review of the fundamentals of com- have plex signal processing. Some important identities are given. Examples are given using complex SFG representations. In the Yt() = yr()t + jyq()t (1) 1. A paper was submitted to the IEEE Transactions on Circuits and Systems, jω t = Xt()e i in 1981, but it was inexplicably rejected; a copy of this key paper submission has been obtained and is available at [8]. third section, a number of classical wireless modulations sys- x y tems are described using the complex formulation. These sys- 1r r X1 Y x tems include Weaver and Hartley modulators and polyphase 2r X2 filters. Errors due to non-ideal coefficients in mixers are quan- x1q yr tified. In the fourth section, complex analog and digital filters x2q are discussed. The fifth section describes more-recent complex (a) (b) signal-processing applications. These include complex adap- Fig. 2. (a) The real SFG for adding two complex signals and tive image-reject mixers, complex low-IF receiver architec- (b) the equivalent complex SFG. tures including a new modulation approach that doubles the data rate for a given A/D sampling frequency, and complex a x = cos()ω t yr jω t band-pass over-sampling A/D converters, as examples. These r i e i Y systems are especially useful in multi-standard wireless appli- b cations as they are inherently wide-band (without excessive b Kajb= + imaging errors) and allow the systems to be adapted to individ- (b) xq = sin()ωit yq ual standards using digital programming after the A/D convert- a (a) ers. 2. Complex Signal Processing Fig. 3. (a) The real SFG of a complex multiply and (b) the equivalent complex SFG. Complex signal processing systems operate on ordered-pairs again the complex SFG of a multiplication operation, as shown of real signals. The typical definition of a complex inner-prod- in Fig. 3. (b), is considerably simpler. Note that the real SFG uct is does not include the constant j anywhere, whereas in the com- t plex SFG, it is admissible. 1 ∗ 〈〉x1()t , x2()t = lim -----x1()xτ 2 ()τ dτ (3) t → ∞ 2t∫ A special case of multiplication is when multiplying an –t input signal by the constant j. This is shown using a complex SFG in Fig. 4. (a) and a real SFG in Fig. 4. (b). It is seen this Combining this definition with the typical definitions for com- operation is simply an interchanging of the real and inverted plex inversion, addition and multiplication, and the properties imaginary parts; this operation is the basis behind realizing of commutation and association, constitutes a Hilbert Space. It imaginary components in complex networks. is assumed that all signals are effectively of finite-energy and x y frequency and time limited (or periodic). This implies that a X Y r –1 r signal, xi()t can be represented as a finite series of complex- oids2 j xq yq N (a) (b) jω0it Fig. 4. Multiplying a signal by j . xi()t = ∑ kie (4) iN= – Another important operation is complex conjugation. Con- sider where ki is in general complex [5]. In this paper, we will limit our consideration normally to a single (or a few) complexoids Xt()= Mejφejωit (6) to simplify explanations. The generalization to signals as We have defined by (4) is left for a later date. –jω t 2.1 Complex Operations X∗()t = Me–jφe i (7) A complex addition of two complex signals simply adds the and we see that besides being phase rotated, X has been trans- two real parts and the two imaginary parts independently. Con- formed from a positive frequency into a negative frequency sider the real SFG that represents adding a complex signal to a complexoid. Conjugation also changes negative frequencies second complex signal as shown in Fig. 2. (a). Its complex complexoids to positive frequencies. The conjugation opera- SFG is shown in Fig. 2. (b). tion is shown in the real SFG of Fig. 5. (a). Alternative com- Slightly more complicated is a complex multiplication; con- plex SFG’s are shown in Fig. 5. (b) and (c). sider multiplying a complexoid Xt()= ejωit by a complex coefficient Kajb= + . We have xr yr X conj{ } Y X Y Yt()= {} acos()ωit – bsin()ωit (5) xq yq –1 (b) + ja{}sin()ωit + bcos()ωit –1 (c) (a) The real SFG of this operation is shown in Fig. 3. (a). Once 2. Historically, a complexoid had been called a cissoid. The author prefers the Fig. 5. The complex conjugation operation. former name. Two additional operations are required for realizing complex filters, namely the delay operator for digital filters and the X Y integration operator for analog filters. These operations simply consist of independently applying the real operation to each –jsgn()ω 12⁄ part of the complex signal as shown in Fig. 6. Hilbert Transform (a) j x x r yr r z–1 yr X z–1 Y 12⁄ xq yq xq –1 y z q Hilbert Transform (a) 12⁄ –jsgn()ω –1 xr 1s⁄ yr X 1s⁄ Y –jsgn()ω xq 1s⁄ yq (b) (b) Fig. 7. One possibility for realizing a positive-pass filter Fig. 6. (a) The digital delay operator and (b) the analog (PPF): (a) a complex SFG and (b) an equivalent real SFG.

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