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Low Noise Amplifier Design for Ultra-WideBand Radio Jongrit Lerdworatawee, Won Namgoong Department of Electrical Engineering University of Southern California lerdwora@usc.edu, namgoong@usc.edu Unlike the narrowband LNA, the signal bandwidth of an UWB radio is several orders of magnitude greater. Hence, the ABSTRACT underlying single tone signal assumption employed in narrow- band LNA design becomes invalid, making many of the existing A new theoretical approach for designing a low-noise ampli- narrowband design techniques [4] based on this assumption also fier (LNA) for the ultra-wideband (UWB) radio is presented. unsuitable. Unlike narrowband systems, the use of the noise figure (NF) per- formance metric becomes problematic in UWB systems because The LNA is often designed to present an input impedence of of the difficulty in defining the signal-to-noise ratio (SNR). By 50Ω to avoid reflections on the transmission line connecting the defining the SNR as the matched filter bound (MFB), the NF mea- off-chip antenna to the on-chip LNA. In this paper we consider a sures the degree of degradation caused by the LNA in the achiev- highly integrated UWB radio system. We assume that the antenna able receiver performance after the digital decoding process. The is placed in close proximity of the LNA, allowing us to ignore the optimum matching network that minimizes the NF as defined 50Ω input impedence requirement. above has been solved. Since realizing the optimum matching net- For the noise factor (NF, or noise figure in dB) of the LNA to work is in general difficult, an approach for designing a practical be a meaningful metric in an UWB receiver, we define the SNR as but suboptimum matching network is also presented. The NF per- the matched filter bound (MFB) [5], which represents an upper formance of both the optimum and the suboptimum matching net- limit on the performance of data transmission systems. The MFB works is studied as a function of the LNA gain. is obtained when a noise whitened matched filter is employed to receive a single transmitted pulse. By defining the SNR as the MFB, the NF measures the degree of degradation caused by the LNA in the achievable receiver performance after the eventual 1 INTRODUCTION digital decoding process. In this paper, the optimum matching net- The ultra-wideband (UWB) radio is a relatively new technol- work that minimizes the NF as defined above has been solved. ogy that is being pursued for both commercial and military pur- Since the optimal LNA matching network is generally difficult to poses [1][2]. It operates by spreading the energy of the radio realize in practice, we also present an approach for designing a signal very thinly over a wide bandwidth (e.g. several gigahertz). sub-optimal but practical matching network. The rationale for deploying the UWB radio systems lies in the The paper is organized as follows. The circuit and system benefits of exceptionally wide bandwidths, thereby achieving a model of the LNA-Antenna is presented in Section 2. In Section 3, combination of very fine time/range resolution, ability to resolve the general solution to the optimal and suboptimal matching LNA multipath components, and favorable propagation condition of are derived. Performance results are presented in Section 4. Con- material penetration at low frequencies [3]. clusions are drawn in Section 5. The goal of the receiver analog front-end is to condition the received analog signal for digitiziation, so that the highest perfor- 2 CIRCUIT AND SYSTEM MODEL mance can be achieved after decoding in the digital domain. The first and probably the most critical component of the analog front- Throughout this paper, capital letters are used to denote the end is the low noise amplifier (LNA), whose purpose is to amplify Fourier transforms (e.g. X(ω)) of (voltage or current) system the received signal from the antenna with as little distortion and responses in the time domain, which are written in the correspond- additional noise as possible. This is achieved by designing an ing lower case letters (e.g. x(t)). Sometimes the terms ω and t are appropriate matching network placed between the antenna and the omitted for notational brevity unless needed for clarity. amplifier. 2.1 Circuit model of LNA-Antenna This work was supported in part by the Army Research Office The quasi-static MOS transistor model is employed in this under contract number DAAD19-01-1-0477 and National Sci- paper to account for the high-field effects in short-channel devices ence Foundation under contract number ECS-0134629. [6]. Accordingly, the transconductance gm and the gate-source capacitance Cgs can be represented in terms of the power dissipa- where S i i ( ω ) is the PSD of the cross-correlation of ig(t) and g d tion Po (= IdVsupply) and the normalized gate overdrive ρ (= (Vgs- id(t). For a long channel device c = 0.395j. For lack of a more Vth)/Lεsat): accurate value currently available, we assume that the long chan- 2P o 1+ρ⁄2 nel values for c, α, γ and δ are also valid in the short channel - g m = ------------------------------ -------------------- (1) model employed in this paper. V supply Lεsat ρ ( 1 + ρ ) Po 1+ρ C gs = -- ----------------------------------- ----------- 2 2.2 System model of LNA-Antenna - - - (2) 3 V supply v sat ε sat ρ 2 where L is the gate length, Vgs is the gate-source bias voltage, Vth jXb ig(t) Rs+jXa gmZload vo(t) is the threshold voltage, vsat and εsat are the saturation velocity Rs+j(Xa+Xb) and electric field, respectively. vs(t) id(t)/gm jXa(ω) v(t) vs(t) Rs(ω) jXs(ω) jX1(ω) ig(t) gmVgs id(t) Zload Figure 2 : System block diagram of LNA. + + + v(t) + Vgs vo(t) Fig. 2 is the system model of the circuit model in Fig. 1. The source _ _ impedance objective is to design the causal matching network (i.e., Xa(ω) jX2(ω) jωCgs and Xb(ω)) so that the SNR at the output vo(t) is maximized. In matching network jXb(ω) the presence of ig(t), there exists an optimum gain in the matching network that balances the combined effects of ig(t), id(t) and sig- Figure 1 : Circuit model of LNA nal amplification. Fig. 1 shows a circuit model of the analog front-end, includ- Note that ig(t) can be decomposed into two orthogonal com- ing the antenna, the matching network, the LNA and a load, with ponents, i.e., three noise sources: the thermal voltage noise from the antenna resistance vs(t), the MOS gate current noise ig(t), and drain cur- id( t ) i g ( t ) = i gu ( t ) + y c ( t ) ⊗ ---------- - (7) rent noise id(t). With no loss in generality, the antenna is modeled gm as a voltage source v(t) with an impedance Zs(ω) = Rs(ω)+jXs(ω) where ⊗ is the convolution operator, igu(t) is the uncorrelated while the amplifier is assumed as a common-source MOS transis- component of ig(t) to id(t), and yc(t) is the equivalent correlation tor. The matching network is assumed lossless, consisting of two admittance between ig(t) and id(t)/gm. From (4)-(7), the Fourier reactances, X1(ω) and X2(ω), as illustrated by the solid-line block transform of yc(t) (i.e., Yc(ω)) can be obtained and given by in Fig. 1. For ease of analysis, the source reactance Xs(ω) is grouped with X1(ω) and referred to as Xa(ω), and the gate-source Si i ( ω ) δ capacitance Cgs is grouped with X2(ω) and referred to as Xb(ω). Y c ( ω ) = g m ------------------ = jωC gs ⋅ α c g d - ----- Si ( ω ) 5γ (8) d The power spectral density (PSD) of the thermal noise from = jX c ( ω ) the antenna, the drain and the gate noise are given by Using the definition of the SNR described earlier, the SNR S v ( ω ) = 4kTRs ( ω ) (3) s at the input of the LNA (i.e., the SNR of the received input signal) γ is given by [5] S i ( ω ) = 4kT -- g m - (4) d α 2 2 A P( ω) ( ωCgs ) 2 SNR in = ∫ ------------------------ dω Sv ( ω ) - (9) S i ( ω ) = 4kTδα ------------------- - (5) s g 5g m where P(ω) is the normalized channel response to a single multi- -23 where k = 1.38 x 10 J/K is the Boltzmann constant, T is the path component (i.e., ||p||2=1) and A is the scaling factor of the absolute temperature, α is the ratio of gm to the zero-bias drain received signal. With no loss in generality, we assume that the conductance, γ and δ are the coefficients of channel and induced received signal is a 2nd derivative of a Gaussian pulse [2]. gate noise. Random noise process ig(t) is correlated to id(t) with a Similarly, the SNR at the output of the LNA is correlation coefficient c as given by [7] 2 2 A P( ω) Si i ( ω ) g d c = ------------------------------------------- - (6) SNR out = ∫ ------------------------ dω Sn ( ω ) - (10) out Si ( ω ) ⋅ S i ( ω ) g d where S n ( ω ) represents the input-referred noise of all the noise As will be shown in the following sections, a trade-off between out high Gopt and low NFopt exists by varying ρ for a given Po. sources in the LNA, which is given by Note that Xa,opt(ω) and Xb,opt(ω) given in (13) minimize the 2 2 Sn = S v + S i ( Rs + X a ) (11) degradation in SNRout caused by the additive noise at every fre- out s gu quency, and hence they become independent of the received signal 2 2 Si + R s ----- -X c + Xa ----- -X c + 1 ------ 2 1 1 d pulse. In a realistic matching network with a fixed structure, how- - - Xb Xb g2 ever, designing Xa(ω) and Xb(ω) with arbitrary reactances as m where the spectrum of igu(t) is assumed in the optimum matching network is in general not possi- ble. The matching network then becomes a function of the transmit- 2 ( ωC gs ) 2 ted signal pulse. S i ( ω ) = 4kTδα ------------------- ( 1- c ) - (12) gu 5g m 3.2 Suboptimal matching 3 LNA NOISE MATCHING 3.1 Optimal matching Reactances Xa(ω) and Xb(ω) that maximizes SNRout are obtained by differentiating (10) with respect to Xa(ω) and Xb(ω) and setting the result to zero. Assuming the output noise power of igu(t) 2 2 is less than that of id(t) (i.e., S i ( ω )Rs ( ω )g m < S i ( ω ) ), which is gu d typically the case, the optimum Xa(ω) and Xb(ω) (denoted as Xa,opt(ω) and Xb,opt(ω)) can be solved: Xa, opt ( ω ) = ± R s ( ω ) ( 1 ⁄ Γ ( ω ) -R s ( ω ) ) 1 Xb, opt ( ω ) = ------------------------------------------------------------------------------------- (13) − Γ( ω) Xc ( ω ) + -------------- ( 1 ⁄ Γ ( ω ) -Rs ( ω ) ) - Rs ( ω ) Figure 3 : suboptmal vs. optimal reactance in a signal region given gm=1mS and Cgs=1pF where Γ(ω) is Si ( ω ) δα 2 2 2 Since realizing the optimum matching network is in general Γ ( ω ) = ------------------------- = -------- ( 1- c ) ( ωCgs ) gu 2 - - (14) difficult, a heurisitc approach for determining a practical but subop- Si ( ω ) ⁄ g m 5γ d timum matching network is presented. Based on Xa,opt(ω) and Substituting (13) into (11), the optimum NF, denoted as NFopt, is Xb,opt(ω), a structure for the suboptimum matching network that best approximates the optimal response is first selected. The ∫ P ( ω ) dω - 2 antenna impedance is assumed to be 50 Ω across the bandwidth of NF opt = ----------------------------------------------------------- 2 (15) interest. As shown in Fig. 3, the optimal matching network can be P(ω ) ∫ ------------------------------------------------- dω ω 4δγ 2 approximated by a two element (Lm and Cm) L-matching network, i.e., Xa(ω) = jωLa and Xb(ω) = 1/jωCb, where Cb = Cgs + Cm and La 1 + ------ -------- ( 1- c ) - ωT 5 = Lm. The choice of La and Cb is determined by numerically solving where ωT is the unity gain angular frequency: the following constrained optimization problem: ∫ P ( ω ) dω - 2 gm 3 ν sat ( 1 + ρ ⁄ 2 )ρ ω T = ------- = -- -------- --------------------------- - - - - (16) minimize NF = --------------------------------------------------------- (18) Cgs 4 L (1 + ρ)2 2 P(ω) Assuming a resistive load Zload (= Rload), the corresponding ∫ ---------------------------------------------- dω 1 + F1( ω ) + F2( ω ) signal voltage power gain (in units of V2/V2) of the LNA, denoted subject to L a ≥ 0, Cb ≥ C gs (19) as Gopt, is given by where 2 Xb( ω ) δα ( ωCgs ) 2 g m R load ∫ ---------------------------------------------------------------------------- P ( ω ) dω 2 2 2 - 2 2 2 2 F1 ( ω ) = --------------------------- [ R s + ( ωL a ) ] - (20) R s + ( X a, opt ( ω ) + Xb, opt ( ω ) ) 5g m R s G opt - = ------------------------------------------------------------------------------------------------------------------------------- (17) ∫ 2 P ( ω ) dω dB, increasing Po by 20 mW from 10 mW to 30 mW improves G γ 2 2 δ 2 F2 ( ω ) = ---------------- R s ω C b + C gs α c ----- (21) by less than 1.5 dB; whereas increasing Po by only 6 mW from 4 αg m R s 5γ mW to 10 mW increases G by more than 11 dB. This diminishing 2 δ returns in G suggests that large signal amplification is most effi- + 1-ω L a Cb C b + ωC gs α c ----- 2 5γ ciently achieved in multiple stages. The cost function (i.e., NF) in (18) is obtained by substituting Xa(ω) and Xb(ω) of the L-matching network into (10) and (11). 5 CONCLUSIONS Similar to (17), the corresponding signal voltage power gain (in A generalized approach for designing an UWB LNA that units of V2/V2) is minimizes the NF with the SNR defined using the MFB has been developed. The matching network consists of two lossless reac- 2 P(ω ) tances, which are connected in series and in parallel to the MOS g m R load ∫ --------------------------------------------------------------- dω 2 2 2 2 amplifier. The optimum matching network depends only on the ( 1-ω 2 L a C b ) + ( ωR s C b ) G = ----------------------------------------------------------------------------------------------- (22) LNA device noise while the suboptimum matching network ∫ P ( ω ) dω 2 depends also on the received signal and noise. This additional dependency of the suboptimum matching network results because 4 PERFORMANCE RESULTS designing matching networks with arbitrary reactances as assumed in the optimum matching network is in general not pos- sible. For a simple LC suboptimum matching network that we con- sidered, there exists an optimum G that minimizes the NF for a given power dissipation. Although the optimum G can be increased by increasing power consumption, this approach suffers from diminishing returns. Hence, a single stage amplification may not be sufficient; more complex matching network or multiple amplification stages with attendant complexity may be required. REFERENCES [1] M. Z. Win and R. A. Scholtz, “Impulse radio: How it works,” IEEE Commun. Lett., vol. 2, no. 2, pp.36-38, Feb. 1998. [2] M. Z. Win and R. A. Scholtz, “Ultra-wide bandwidth time- hopping spread-spectrum impulse radio for wireless multi- access communications,” IEEE Trans. Commnu., vol. 48, Figure 4 : Contours of noise figure and signal gain relating ρ for no. 4, pp. 679-691, Apr. 2000. a specified Po (the number on the curves) [3] M. Z. Win and R. A. Scholtz, “On the roubustness of ultrawide bandwidth signals in dense multipath In Fig. 4 the NF is plotted against the signal voltage gain G environments,” IEEE Commun. Lett., vol. 2, no. 2, pp. 51- (in units of V2/V2) for both the optimum (in dash-line) and the 53, Feb. 1998. suboptimum matching networks (in solid-line) when Po is fixed. [4] D. K. Shaeffer, T. H. Lee, “A 1.5-V, 1.5-Hz CMOS low noise amplifier,” IEEE J. Solid-State Circuits, vol. 32, pp. In the optimum matching network, a trade-off between reducing 745-759, May. 1997. NF and increasing G can be made by varying the normalized gate [5] J. Cioffi, “EE379A Course Notes,” Stanford University. overdrive ρ. For sufficiently high G, large increases in G causes [6] K.-Y. Toh, P.-K. Ko, and R. G. Meyer, “An Engineering only a small increase in NF. For example, increasing G from 10 model for short-channel mos devices,” IEEE J. Solid-Sate dB to 20 dB when Po is 10 mW increases the NF by less than 1 Circuits, vol. 23, no. 4, pp. 950-958, Aug. 1988. dB. In the suboptimum matching network, there is an optimum G [7] D. P. Triantis, A. N. Birbas, and D. Kondis, “Thermal noise that minimizes the NF. For example, when Po is 10 mW, the opti- modeling for short-channel MOSFET’s,” IEEE Trans. mum G is approximately 7 dB. If operating below the optimum G, Electron Devices, vol. 43, pp. 1950-1955, Nov. 1996. the NF does not increase much. However, if G is increased beyond the optimum point, the NF increases abruptly. For exam- ple, an increase in G from 10 dB to 20 dB when Po is 10 mW increases the NF by almost 10 dB. Hence, the LNA that dissipates Po should not be designed to operate with a gain that is much greater than the optimum G. Another important observation is that for a fixed NF, increasing G by increasing Po suffers from dimin- ishing returns. For example, given a target NF of approximately 3