High-Frequency Noise in RF Active CMOS Mixers

Payam Heydari Department of EECS University of California, Irvine Irvine 92697-2625

Abstract - A new analytical model for high-frequency noise pair on the output noise and the NF. Simulation results are pro- in RF active CMOS mixers such as single-balanced and dou- vided in Section 4. ble-balanced architectures is presented. The analysis includes the contribution of non-white gate-induced noise at 2. BACKGROUND the output as well as the spot noise figure (NF) of the RF CMOS mixer, while accounting for the non-zero correlation A. Large-Signal Analytical Model for RF CMOS Mixers between the gate-induced noise and the channel thermal noise. The noise contribution of the RF transconductor as Figs 1 (a) and (b) depict the double-balanced and single-bal- well as the switching pair on the output noise is discussed. anced cells, respectively, that are extensively used to implement The analytical model predicts that the output noise and NF active switching mixers. are both a strong function of the LO frequency at gigahertz To achieve an accurate analytical model for an RF CMOS range of frequencies. Simulation results verify the accuracy mixer in nanometer technology, the I-V characteristics of a of the analytical model. short-channel device is utilized [4]:

1. INTRODUCTION I I ID1 ID2 D3 D4 Mixers in radio-frequency (RF) transceivers are considered MN1 MN2 MN3 MN4 as nonlinear blocks. The frequency-varying bias-points of active VLO switching devices due to the large-signal behavior of mixers cause the mixer noise to become essentially a cyclostationary − MN MN5 ISS+is ISS is 6 stochastic process. In addition, the wireless standards such as Vin bluetooth and IEEE802.11b operating in multi-gigahertz range (a) demand front-end circuits including the low-noise amplifier 2ISS (LNA) and the mixer to operate at giga-hertz range of frequen- cies. The low-frequency analytical models for the mixer noise proposed in [1] and [2] are, therefore, incapable of accurately ID1 ID2 predicting the noise characteristics of the mixer such as the noise figure (NF), the output noise, or the input-referred noise. More precisely, at RF frequencies, the random potential fluctua- MN1 MN2 tions in the conducting channel of a MOS transistor are coupled VLO to the gate terminal through the oxide capacitance and cause a MN3 ISS+is gate noise current even all terminal voltages of the MOS device Vin are fixed. Known as gate-induced noise, this noise source is cor- (b) related to the channel thermal noise and its power spectral den- sity (PSD) increases with frequency, hence it is a non-white Fig. 1. Active blocks used in switching mixers, (a) a CMOS Gilbert random process. [3] proposed a stochastic differential equation cell. (b) a differential pair. approach to characterize the noise in CMOS switching mixers. ()V – V 2 [3] was able to consider the cyclostationary noise generation in υ GS TH mixers. However, this model cannot efficiently analyze the ID = W sat Cox ------(1) VGS – VTH + LEsat noise in an active mixer in which the switching pair is mostly where W and L are the transistor channel width and length, operating in the triode region whereas the RF transconductor ν operates in the saturation region. respectively, Cox is the gate-oxide capacitance, sat is the satu- rated velocity, and Esat is the saturated electric field. Using Eq. In this paper we present a high-frequency noise model for the (1), the ratio of the instantaneous transconductance g of a active CMOS mixers while considering the short-channel m short-channel transistor in terms of its DC bias current IDQ is: effects in MOS devices in nanometer technologies. The pro- ()⁄ posed analytical model is simplified to the noise model pre- 2IDQ VGS – VTH 2 + LEsat gm = ------(2) sented in [1] at low frequencies. Closed-form expressions VGS – VTH VGS – VTH + LEsat resulting from the analytical model gives circuit designers an understanding of how the high-frequency noise affects the Since a large LO signal is applied to the switching transis- mixer NF and output noise. tors, the bias points of MN1 and MN2 are not fixed but vary periodically [1]. The output current of the single-balanced mixer Section 2 briefly reviews single and double-balanced mixers, shown in Fig. 1 (b) is a function of the instantaneous LO voltage and the dominant device noise sources in high-frequencies. Sec- V and the current at the output of the tail current I +i . tion 3 presents the noise analysis of the CMOS mixer and illus- LO SS s trates the noise contribution of the transconductor and switching Therefore, we have: ∂FV(), I cyclostationary stochastic process. In a cyclostationary process, (), (), LO SS Io = ID1– ID2 = FVLO ISS + is = FVLO ISS + ------∂()-is the mean and autocorrelation function are both periodic. As a ISS + is consequence, the power spectral density (PSD) is time-varying. or () () There are three major noise sources that contribute to the Io = p0 t + p1 t is (3) where p (t) and p (t) are two periodic waveforms [1]. In a Gil- noise at the output of the mixer; noise from the transconductor, 0 1 noise from the switching stage, and noise coming from LO. bert-cell double-balanced mixer, p0(t) is eliminated because of Note that each of these three major noise sources can be a com- the presence of two stages of differential signaling. When the bination of constituent device noise sources. A comprehensive absolute value of the LO voltage is less than a certain voltage, high-frequency noise analysis of the RF CMOS mixer in the Vx, the tail current is shared between the two switching devices presence of the three noise sources is given in the following. MN1 and MN2. Otherwise, only one stage turns on and the Throughout the forthcoming analysis, we assume that ∆f = 1Hz absolute value of p1(t) thus becomes one, as shown in Fig. 2. to simplify the noise expressions. v (t) LO Vm Vx A. Noise from the RF Transconductor Fig. 3 (a) shows a single-balanced cell along with the input- − t Vx referred noise sources due to the noise contribution of MN3 at − ISS Vm the input of the transconductor. From a system-level perspec- p0(t) tive, the noise signal at the transconductor is multiplied by the t switching pair’s instantaneous current gain p1(t) and generates a − I 1 SS current noise io3(t), as shown in Fig. 3 (b). p1(t) ID1 ID2 t −1 MN MN ∆ 1 2 p1(t) io3(t) VLO Fig. 2. p0(t) and p1(t) waveforms [1]. n (t) MN 3 B. High-Frequency Noise in MOSFETs 3 V 2 2 (a) (b) The ohmic regions of a MOSFET exhibit finite resistivity, n3 In3 thereby contributing thermal noise. In RF integrated circuits, Fig. 3. (a) Noise contribution of the transconductor in a single bal- devices with very large channel widths are used. In such cases, anced mixer, (b) mixer operation for the transconductor’s noise. the source and drain ohmic resistances are negligible whereas the gate resistance can be significant. The gate resistance forms i ()t = n ()t p ()t (9) a distributed RC circuit. However, it can be shown that the dis- o3 3 1 tributed effect of the gate resistance in the noise model can be io3(t) is thus a cyclostationary process due to the fact that approximated using a lumped resistance whose value is one- p1(t) is a periodic function and n3(t) is a wide-sense stationary third of the physical ohmic gate resistance [5]. The thermal process. Subsequently, the output noise PSD of the mixer due to noise due to the ohmic poly resistance is thus as follows: the noise from the transconductor is a time-varying function. r However, the time-average of the output noise i (t) is a WSS V 2 = 4KT-----G (5) o3 nrg 3 process [7]. The PSD of this time-average process is [1]: In addition, the thermal fluctuations of channel charge in the ∞ o ()ω, 2 ()ω ω MOSFET produce effects that are modeled by drain and gate Sn3 t = ∑ p1,n Sn3 – n LO (10) current noise sources. These current noise sources are partially n = –∞ correlated with each other, because they share a common physi- 2 Sn3(ω) (which is equal to in3 for ∆f = 1Hz) is the PSD of the cal origin and thus possess a spectral power given by: output current noise generated in MN3 whose value is yet to be 2 γ determined. ind = 4KT gm (6) To derive Sn3(ω), the impact of various noise sources must be 2 δ ing = 4KT gg (7) accounted for. The noise sources contributing on the output i i ∗ noise contain two correlated thermal noise sources, the drain c = ------ng nd- (8) current noise and the gate-induced noise; and uncorrelated noise 2 2 sources containing the thermal noise of ohmic polysilicon gate, ingind the thermal noise of input source resistance, and the flicker ζω()2 2 ⁄ where gg = CGS gm is the real-part of gate-to-source noise. Fig. 4 depicts the high-frequency small-signal model admittance, and γ, δ, and c are bias-dependent factors. For short- including all these noise sources: channel MOSFETs, γ, and δ are 2.5, and 3, respectively. 2 Because the value of the correlation coefficient, c, in the short- Vn3, 1⁄ f channel regime is not known at present, we will assume that c rG3 R V 2 D remains at its long-channel value, which is j0.395 [6]. Rs3 s3 nrg3 gg3 MN3 CGS3 i2 3. NOISE ANALYSIS V 2 ng3 2 nRs3 ind3 The output noise generated in a mixer has a periodically gm3VGS ω2 2 S time-varying statistics. This is because an active mixer with a CGS3 gg3 = ------ζ - periodic LO is essentially a nonlinear periodic system and its gm3 response to an input stationary noise source is a non-stationary Fig. 4. Noise contribution of the transconductor. process whose statistical properties are invariant to the time- shift by integer multiples of TLO, hence it is modeled as a To quantify the effect of noise at drain node of MN3, the con- The noise contributions of correlated current noise sources, tribution of uncorrelated noise sources are first considered sepa- ing3,c and ind3 on the overall PSD of the drain current of MN3 are rately. The overall output noise due uncorrelated noise sources is determined by obtaining the current response to each individual then calculated using superposition. The three uncorrelated noise current noise source, and then computing the mean-square of the sources in Fig. 4 are thermal noise of ohmic polysilicon gate rG3 , resulting summation of these two individual responses. The the thermal noise of input source resistance Rs3, and the flicker channel noise source ind3 directly appears at drain node of MN3. noise. The noise at the output of MN3 due to each uncorrelated As for ing3,c, first, the output current of MN3 in response to this noise source is obtained by multiplying its PSD with the square current source is obtained. ing3,c is fully correlated with ind3. of the transfer function. Table 1 summarizes the derivations for Therefore, it is treated as a current source proportional to i , thermal noise of ohmic polysilicon gate and the thermal noise of nd3 input source resistance. In the derivation of expressions in Table i.e., ⁄ ()ζω2 2 jωC δζ jωC 1, it is assumed that (rG3+Rs3) << gm3 CGS3 . {} GS3 {}κ GS3 {} F ing3, c ==------c ------F ind3 c ------F ind3 (15) gm3 γ gm3 Table 1: Input noise sources and the corresponding PSD of the where F{.} represents the Fourier transformation. The total noise noise responses at the output of MN3. current at the drain of MN3 due to ing3,c and ind3 is readily found using small-signal high-frequency circuit for the transistor shown Input noise source PSD of the noise at the output in Fig. 4:

g2 2 2 V 2 2 m 3 []()κ κ ()() nrg3 Vnrg3 ------inc, MN3 = Fc + 1 ind3 + c ind3 hMN3 t (16) []ωC ()r + R 2 + 1 * GS3 G 3 s3  2 g2 V m3 () ⁄ τ ( ⁄ τ ) τ nRs3 V 2 ------where hMN3 t = 1 G3exp –t G3 , and G3 = CGS3RsG3 . Also, nRs3 []ωC ()r + R 2 + 1 GS3 G 3 s3 x(t)*y(t) stands for the convolution of x and y. Using stochastic 2 analysis (the details are omitted due to the lack of space), the The overall output noise due to the noise sources, Vnrg3 and 2 overall PSD of the current noise due to the correlated current VnRs3 , is frequency-dependent, which, in turn, results in a non- sources ing3,c and i is as follows: white noise at the output of the transistor. To simplify the analyt- nd3 ()κ 2ω2 2 2 ical model, we obtain an equivalent white noise process for this 1 + c + 1 CGS3RsG3 2 S ()ω = ------ind3 (17) colored-noise process at the output by averaging the noise in the n3 i ,i ω2 2 2 frequency domain over the LO frequency: ng3, c nd3 1 + CGS3RsG3 f Combining (13) and (17) yields the overall PSD of partially cor- 2 2 LO 2 ()Vnrg3 + VnRs3 g related noise sources ing3 and ind3: u ()ω ≈ ------m3 - df (11) Sn3 ∫ 2 fLO []ωC ()r + R + 1 ()ω 0 GS3 G3 s3 Sn3 = i , i Calculating the above integral, the overall noise due to the ng3 nd3 uncorrelated noise sources, therefore, is as follows: 1 2 2 1 + ()κ + 1 2 + κ2------– 1 ω2 C R ω –1 ω c cc 2 GS3 sG3 S u ()ω = 4KTg2 R′ ------p3 tan ------LO (12) ------4KTγg n3 m3 sG3 ω ω ω2 2 2 m3 LO p3 1 + CGS3RsG3 ω ≅ ()–1 where R’ sG3 = rG3/3+Rs3, RsG3 = rG3+Rs3, and p3 CGS3 RsG3 . For the LO frequency much smaller than ω0, (12) is simplified to (18) u ()ω 2 ′ A plot of S ()ω , vs. frequency is depicted in Fig. 5. Sn3 = 4KTgm3RsG3 . n3 ing3 ind3 As for the flicker noise, it appears at the output of the mixer ()ω []ω2 ⁄ ω2 2 Sn3 , around fLO and all the odd-harmonics. The PSD of the flicker Nss3 = p3 z3 ind3 ing3 ind3 Nss3 noise at the output of the transconductor is found using equations ω2 ⁄ ω2 1+ p3 z3 Np3 similar to (11) and (12) with the consideration that the flicker N = ------i2 p3 1+ ω2 ⁄ ω2 nd3 noise is a colored noise source. z3 p3 Nz3 2 2 To account for the partially correlated noise sources, a more N = ------i 4ΚΤ γg z3 1+ ω2 ⁄ ω2 nd3 m3 complicated analysis needs to be undertaken. Recall that the gate z3 p3 1 induced noise and drain noise are partially correlated stochastic ω2 = ------p3 2 2 ω ω ω C R z3 p3 10ω processes with a correlation coefficient given by (8). This non- GS3 sG3 1 p3 ω2 = ------zero correlation between the noise sources creates an intricacy in z3 1 ()κ + 1 2 + κ2------– 1 C 2 R 2 the derivation of the PSD of the output current noise, because the c cc 2 GS3 sG3 superposition principle is not held valid for the power of corre- Fig. 5. The PSD of S ()ω , vs. the radian frequency lated stochastic processes. To perform the noise analysis of par- n3 ing 3 ind3 tially correlated current noise sources, ing3 and ind3, the gate induced noise is first expressed as a superposition of its corre- Direct substitution of S ()ω , in Eq. (10) yields a periodic n3 ing 3 ind3 lated and uncorrelated components [6], i.e., expansion of S ()ω , with a fundamental period 1/f . n3 ing3 ind3 LO 2 2 2 2 2 However, S ()ω , is frequency-dependent, and hence, a i ==i , + i , 4KTδg ()1 – c + 4KTδg c (13) n3 ing3 ind3 ng3 ng3 u ng3 c g3 g3 direct substitution does not result in a closed-form expression for where c is given by (8). The uncorrelated component, ing3,u , is the output current noise of the transconductor. To alleviate this treated in the conventional way by obtaining the circuit response problem, we find an equivalent white noise process over the LO using the small-signal transfer function of MN3. Assuming ()ω frequency for the colored noise component of S i , i . ⁄ ζω2 2 n3 ng3 nd3 (rG3+Rs3) << gm3 CGS3 , we have: The average power of this equivalent white noise process must g2 R2 be equal to the total area under the colored noise PSD in the ()ω m3 sG3 2 Sn3 = ------ing3, u (14) interval [0, fLO]: ing 3, u ω2 2 2 1 + CGS3 RsG3 ω LO for. Solving the noise equations for the switching pair gives rise c 1 ()ω ω to the following equations: S ()ω = ------ω Sn3 , d n3 LO∫ ing3 ind3 2 0 2 2 2 2  ID1 = I ++I I (24) ∑ D1 R D1 i , i D1 i ω2 ω2 j = 1 sGj ngj, c nd j ng j, u p3 p3 ωp3 –1 ω = 4KTγg ------+ 1 – ------------tan ------LO (19) m3 ω2 ω2 ω ω With the aid of the high-frequency circuit model for the tran- z3 z3 LO p3 sistor, the frequency-average of each term in (24) is:

2 ω2 ω2 ω ω ω ω 2 4KTγ gmigmj p12 p12 p12 –1 LO where p3 and z3, defined in Fig. 5, specify the pole and zero of I = ------------------+ 1 – ------------tan ------D1 , g g +g ω2 ω2 ω ω (25) ()ω , ingi indi mi mi mj z12 z12 LO p12 Sn3 ing ind, respectively. As expected, the gate-induced noise g g 2 ω ω has a high frequency contribution on the overall noise at the drain 2 ′ mi mj p12 –1LO ID1 = 4KTRsG12 ------ω tan ------(26) node of the transistor. Ignoring the gate-induced noise results in R gmi+gmj LO ωp 12 high-frequency error in the noise estimation. In fact, experimen- sG12 tal results given in Section 4, verify this observation. The overall for i, j = {1, 2} and ij≠ PSD of the noise at the output of MN3 is a summation of the out- ω () put noise due to the correlated noise sources and the output noise wherep12 =1⁄ RsG12 CGS12 , 1 due to the uncorrelated noise sources: ω2 = ------z12 1 ()κ + 1 2 + κ2------– 1 C 2 R2 ()ω c ()ω u ()ω c cc 2 GS12 sG12 Sn3 = Sn3 + Sn3 ω2 ω2 Similarly, the noise PSD at I is calculated. The output noise is the p3 p3 ωp3 –1 ω D2 = 4KTg γ------+ g R′ + γ1 – ------------tan ------LO- (20) m3ω2 m3 sG3 ω2 ωLO ω sum of the PSD at I and I , which leads to the following: z3 z3 p3 D1 D2 S o ()ω, t ==I 2 + I 2 As a consequence, Sn3(ω−nωLO) in Eq. (10) becomes fre- n12 D1 D2 o 2 ω2 ()ω, g g  ωp12 2g g ′ p12 ω –1 ω  quency-independent. Sn3 t is thus simplified to: 8KT------m1 m2 γ ------+ ------m1 m2 R + γ 1 – ------------p12 tan ------LO- g +g  ω2 g +g sG 12 ω2 ω ω  ∞ m1 m2  z12 m1 m2 z12 LO p 12  o ()ω, ()ω 2 ()ω Sn3 t ==Sn3 ∑ p1,n Ep1 Sn3 (21) (27) n = – ∞ Recall that gm1 and gm2 are the instantaneous small-signal where Sn3(ω) is given by Eq. (20), and Ep1 is the energy of the transconductances of switching devices MN1 and MN2 in Fig. 1 Fourier series coefficients of the periodic signal p (t) and is (b). gm1 and gm2 are periodic signals with a cycle time of 1/ fLO. 1 () () determined using the Parseval’s relation [8]: The transconductance of the pair, G t = 2gm1gm2 ⁄ gm1 + gm2 , o ()ω, ∞ TLO will thus be a periodic function making Sn12 t to be a peri- 2 1 2 o ()ω, E ==p , ------[]p ()t dt (22) odic function of time. The time-average PSD of Sn12 t gives p1 ∑ 1 n T ∫ 1 n = –∞ LO 0 a time-invariant PSD of the switching pair, i.e., o ()ω, ≅ For large LO amplitudes, p1(t) is a periodic square-wave Sn12 t waveform and the average energy per period becomes one. 2 ω2 2ISS  ωp12 1.16ISS ′ p12 ω –1 ω  4KT------γ ------+ ------R + γ 1 – ------------p12- tan ------LO-  The overall time-average output noise for the single-balanced π  ω2 π sG 12 ω2 ω ω V0  z12 Vx z12 LO p 12  mixer shown in Fig. 1 (b) is given by (21). A double-balanced (28) mixer implemented by the Gilbert-cell, should be noisier than a single-balanced mixer because it uses more transistors. A simple The output noise contribution of the double-balanced mixer is extension of the above analysis to the double-balanced mixer twice the value given by (28). results in the following noise expression for the double-balanced To obtain the flicker noise contribution from the switching mixer of Fig. 1(a): pair, a similar analysis is performed with the consideration that the flicker noise is a colored-noise source. o ()ω, ()ω ()ω 2 Sn56 t ==Ep1 Sn56 2Ep1 Sn3 – 4KTgm3Rs3 (23) The noise at the output of the local oscillator has a cyclo-sta- tionary behavior. This is due to the fact that LO is a periodically B. Noise from the Switching Pair time-varying circuit. The device noise sources in an oscillator are The input to the switching pair of an RF mixer is a large LO directly contributing to the phase and amplitude noise of an oscil- lator [9]. As a simplified treatment, it is assumed that the noise signal. For a proper circuit operation we assume that VLO

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