A Study of Out-of-Band Emission in Digital Transmitters Due to PLL Phase Noise, Circuit Non-Linearity, and Bandwidth Limitation

A thorough investigation of major contributors to out-of-band emission (OOBE) in transmitters (TXs) utilizing digital modulation schemes is provided. Specifically, the paper delves into the detrimental effects of phase noise of the local oscillator (LO), typically realized using a phase-locked loop (PLL), on the OOBE phenomenon. Furthermore, the effects of the circuit nonlinearity in a TX, widely recognized as a primary contributor to spectral regrowth and elevated levels of OOBE, are investigated. Additionally, the impact of filtering and bandwidth (BW) limitation on OOBE is taken into account. Comprehensive simulations verify the accuracy of the analytical study. The results provided throughout this paper can be used to determine the linearity and phase noise requirements of different blocks, such as PLL and power amplifier (PA) within a TX chain to design a system complying with a specific mask emission dictated by a particular standard.


I. INTRODUCTION
T HE INTERCONNECTED nature of today's world has necessitated the implementation of a multitude of standards for various wireless communication technologies.This has resulted in a densely populated spectrum, with different frequency bands allocated for various applications and numerous channels within each specific band.With the increasing number of devices and technologies vying for the limited frequency band, it has become increasingly challenging to maintain spectral purity and control out-of-band emission (OOBE), which is defined as undesirable signals appearing at frequencies outside the intended transmission band, potentially causing interference with other communication systems.The interference with other communication systems, directly caused by OOBE, results in a degradation of the received-signal quality [1], [2], [3].Accordingly, significant research efforts have been undertaken to reduce the OOBE power levels associated with different modulation schemes in RF transmitters (TXs) [4], [5], [6], [7], [8].However, TX designs aiming for OOBE reduction may introduce additional distortions that increase the system biterror-rate (BER), resulting in a trade-off between OOBE and BER [9].
PLL, being an omnipresent block in a TX chain, is utilized to generate the LO signal for the upconversion of baseband spectrum to a carrier frequency.Phase noise, an inevitable nonideality associated with a PLL, degrades the performance of transceivers (TRXs) in multiple ways.For instance, previous studies have demonstrated that the PLL phase noise has an adverse impact on the performance of TXs and receivers (RXs), as evidenced by the degradation of errorvector-magnitude (EVM) and bit-error-rate (BER) [10], [11].These unfavorable effects are not exclusive to the in-band signal spectrum.As will be investigated throughout this paper, the PLL phase noise also degrades the signal quality of adjacent-channel TXs.Consequently, it is a critical factor detrimentally affecting the spectral purity of the transmitted signal.This, in turn, gives rise to significant spectral spreading, ultimately leading to elevated levels of OOBE.
In addition to phase noise, another unavoidable circuit nonideality affecting the performance of TRXs is the nonlinearity of various blocks in a TX chain.Circuit nonlinearity, which is characterized by 1-dB compression point (P 1dB ) and n th -order intercept point (IP n ), can result in the generation of harmonics and intermodulation products that may extend beyond the intended frequency band [1], [12].The presence of these harmonics and intermodulation products can generate large in-band blockers due to the circuit nonlinearity, worsening the OOBE.Therefore, various filtering stages are used within a TX to control the spectrum of the transmitted signal and to reduce OOBE levels.
Figure 1 shows the block diagram of a simplified directconversion TX architecture.The baseband quadrature data inputs, x I (t) and x Q (t), are directly upconverted to a pair of RF signals, y I (t) and y Q (t), centered at the carrier frequency, f c , through a pair of I and Q mixers whose other inputs are fed by a PLL generating the LO quadrature signals, g I (t) and g Q (t).The RF quadrature signal, y(t), is obtained by combining y I (t) and y Q (t).Finally, y(t) emerging from the combiner is fed to a PA to boost its power level to a desired value.This amplified signal is denoted by z(t) (Figure 1).
A digital signal processing unit can effectively filter or shape the out-of-band components of the baseband signal while preserving the in-band spectrum.Therefore, the baseband bandwidth is limited to a certain value specified by a specific standard.This bandwidth-limited signal, upon multiplication with an LO signal, creates out-of-band frequency components because of two main reasons: (1) the RF mixing often involves multiplication of the baseband signal with the fundamental component as well as higher harmonics of the LO signal, thereby producing unwanted frequency components at the output of the mixer [13].(2) The LO signal itself is never a pure sinusoid.On the contrary, it always contains phase noise which, after mixing with the baseband signal, produces out-of-band components.The situation is exacerbated when the resulting RF signal passes through nonlinear blocks such as a PA.The signal passing through the PA, due to its non-linearity, experiences AM/AM and AM/PM distortions as well as spectral regrowth, further increasing the OOB power [14], [15].An ideal band-pass filter (BPF) at the PA output should only pass the inband portion of the signal while eliminating its out-of-band components.However, in practice, passive BPFs, due to their limited quality factor, always attenuate the in-band frequency components, and never completely remove the out-of-band portion of the spectrum [16], [17].
The remaining part of this paper is organized, as follows: Section II provides the time and frequency domain representations of the baseband signal.Section III studies the phase noise of integer-N and fractional-N frequency synthesizers and provides approximations to phase-noise expressions to be used in the subsequent sections.Section IV quantifies the impact of LO phase noise on the OOBE of a TX.Section V studies the circuit nonlinearity and bandwidth limitation effects on the OOBE.Section VI presents simulation results of phase noise and nonlinearity impacts on a TX's OOBE, and provides a comparison with the analytical models developed in Sections II-V.Finally, Section VIII provides concluding remarks.
Premise: All calculations in this work apply to doublesided spectra.However, for the sake of simplicity, the results are shown for only the positive frequencies (i.e., f ≥ 0).To account for negative frequencies, the reader merely needs to substitute every f in all equations with its absolute value, |f |.

II. OVERVIEW OF BASEBAND SIGNAL REPRESENTATION
A digital bit can be represented by a pulse transmitted within a specified time window, and any pulse that exceeds beyond this designated timeframe may interfere with adjacent pulses.This phenomenon, commonly known as intersymbol interference (ISI), causes errors in the received signal.The ISI, stemming from channel distortion or imperfect filtering, can introduce errors into the received signal.Therefore, the main goal of advanced transmission techniques incorporating digital modulation schemes, such as quadrature amplitude modulation (QAM) and pulse amplitude modulation (PAM) is to design pulses that minimize ISI and improve the transmission quality.Raised cosine pulse shaping is a common technique used in digital communications to shape transmitted signals into a specific form so as to minimize ISI.In the time domain, the raised-cosine pulse is expressed, as follows [18], [19] where V p and β are amplitude and the roll-off factor of the pulse-shaped signal, respectively, and the signal exhibits zero amplitude at integer multiples of T b .Accordingly, the Fourier transform of a raised-cosine pulse is obtained, as follows [19] The temporal and spectral waveforms of the raised-cosine pulse are shown in Figures 2(a)-2(b), where the spectrum is confined within the bandwidth (defined to be the spectral width of the main lobe) of 1+β 2T b .For β = 0, which is abundantly used throughout this paper, (2) becomes: Suppose the baseband input, x(t), takes the form of a PAM signal where the digital input stream, b k = ±1, ±3, . . ., ±(2 M − 1), directly modulate a raised-cosine pulse, p(t), at a baud rate of 1/T b , i.e., With P(f ) satisfying (3), transmission with zero ISI is possible [19], [20].It can be proved that if b k = ±1, ±3, . . ., ±(2 M − 1) occurs with equal probabilities, the power spectral density (PSD) of x(t) is derived to be [1], [21], [22] where σ 2 b is the variance of the input digital stream, which, for the case of a PAM-2 M signal, is calculated to be S X (f ) for β = 0 is expressed as: where f BB = 1/2T b , and S BB is defined as S BB = (4 M − 1)V 2 p T b /3.Therefore, the total power P X of the baseband signal is calculated to be: Equations ( 7) and ( 8) are used throughout this paper as the PSD and total power of the input baseband signal, respectively.

III. LOCAL OSCILLATOR PHASE NOISE AND SPURS A. INTEGER-N FREQUENCY SYNTHESIZER
This section studies the output noise PSD of a type-II integer-N PLL [1], [23], and investigates the contribution of the charge pump and loop filter non-idealities in generating spurs [10], [24].To accurately capture the underlying mechanisms responsible for phase noise, a linear time-varying (LTV) model [25], [26] has been demonstrated to be an effective mathematical representation, resulting in: where α is a circuit-dependent parameter [27], and γ = αk 1/f 2 dc /2 2 rms , and k 1/f denotes the device 1/f corner frequency, and dc and rms are the DC and RMS values of the impulse sensitivity function (ISF) [25], [26], respectively.Moreover, PLL-based frequency synthesizers for wireless TRXs are often fed by crystal oscillators.A crystal oscillator typically displays a flat phase-noise profile beyond an offset of a few kilohertz [1], [28].Therefore, the phase noise of the input clock is assumed to be white with a double-sided PSD of S I (f ) = η.The VCO and input noise sources are shaped by the loop transfer function from the corresponding source to the output.The PLL transfer function due to VCO phase noise exhibits high pass behavior [23], i.e., where ζ is the damping factor and ω n is the natural frequency: R 1 and C 1 are the loop filter's resistor and capacitor, respectively.I p is the charge-pump current, and N is the division ratio between output and reference frequencies.As for the reference's noise contribution, the transfer function from the input to the output shows a low pass behavior [23], i.e., Leveraging the superposition principle, the PLL's phasenoise PSD is derived to be: Equation ( 13) at low and high frequencies can be approximated, as follows Therefore, ( 13) can be well approximated by: where ). Figure 3 shows plots of ( 13) and ( 16) as well as input and VCO phase noise profiles.This plot verifies the accuracy of this approximation.
In addition to phase noise, an integer-N PLL exhibits spurs at its output due to the charge-pump leakage current, I leak , which draws (injects) current from (to) the PLL loop filter.This current creates a disturbance, V c , on the control voltage, which is derived to be: Under the locking condition, the PLL loop reacts to this disturbance by injecting a large current I p compared to I leak through the charge pump to the loop filter such that the average value of the control voltage remains intact [24].According to Appendix A, the charge pump leakage current as well as the capacitor's finite quality factor produce an effective leakage current, I l,tot , disturbing V c .This phenomenon produces spurs at the output of the PLL separated from the PLL output carrier frequency by integer multiples of the reference frequency (i.e., n × f r ), and their power with respect to the carrier power, P C , is expressed, as follows [10] Assuming the carrier to have unity power, the PSD associated with the carrier waveform and the PLL spurs, downconverted to the baseband, is derived to be: From (18), it is seen that the spurs' power varies in proportion to 1/n 4 .Therefore, the impact of spurs for n ≥ 2 is negligible.Equation ( 19) can thus be approximated by: The output spectrum of a generic integer-N PLL is shown in Figure 4.

B. FRACTIONAL-N FREQUENCY SYNTHESIZER
An integer-N frequency synthesizer is only capable of generating frequencies that are integer multiples of the reference frequency.This constraint creates several critical limitations in designing the PLL for a wireless TRX.For instance, the TX's output channel spacing must be equal to or a multiple integer of the reference frequency of an integer-N PLL.This, in turn, limits the loop bandwidth of the PLL and the extent to which the VCO phase noise is filtered by the loop.Fractional-N frequency synthesizers, on the other hand, overcome this limitation by enabling the TX to have smaller channel spacing relative to the reference frequency.Therefore, the PLL can have higher bandwidth, resulting in a reduction in settling time and phase noise [1], [29].
Figure 5 depicts the block diagram of a fractional-N frequency synthesizer incorporating a modulator for modulus randomization.The modulator produces a binary random modulus control signal, denoted as b(t), which assumes a binary state of either 0 or 1.Three predominant noise sources contribute to the overall noise performance of a fractional-N synthesizer, namely: (1) reference-clock phase noise, (2) VCO phase noise, and (3) quantization noise arising from the random modulus control signal.The reference and VCO phase noise contributions to the PLL output phase noise were investigated in Section III-A.As for the quantization noise contribution, defining B as B = E[b(t)] (where E denotes the expected value of a random variable and 0 ≤ B ≤ 1), the instantaneous frequency of the divider's output is: where q(t) represents the zero-mean quantization noise.If q(t) N + B, (21) can be well-approximated by: Given that the phase transfer functions from the divider output and the reference input to the PLL output are identical, the quantization noise is considered as the excess phase noise of the reference.This excess phase-noise PSD is given by [1]: S q (f ) is the PSD of q(t), which is expressed by ( 24) [1]: where H(f ) is the transfer function of the noise shaping function and sinc(x) = sin(π x)/(π x) [30].In the case where there is no noise shaping function, H(f ) = 1, and for an L th -order noise shaping function, H(f ) is: Since the quantization noise and the reference clock phase noise are independent, the noise PSD due to the quantization noise referred back to the reference clock can be added to the reference clock original phase noise to calculate the total input phase noise, which is derived to be: Since the high-frequency components of ( 26) are filtered by the loop, only the low-frequency components are important.For f f r , ( 26) is simplified to: where λ L is defined as the quantization noise factor (QNF). λ L represents a measure of the quantization-noise impact on the PLL phase noise and is derived to be It is readily observed that QNF is a positive quantity, and in the presence of noise shaping (i.e., L ≥ 1), monotonically decreases with the reference frequency and assumes a peak at B = 0.5.On the other hand, in the absence of noise shaping, QNF increases with the clock reference frequency.Figures 6(a To quantify the impact of quantization noise on the PLL's output phase-noise PSD, every η in ( 13) is replaced by S I (f ) in (26).S (f ) at low and high frequencies becomes:

S (f )
where S R = (N + B) 2 η captures the contribution of the reference phase noise to the PLL output.S (f ) is thus approximated by: where f 2 is found by solving the following equation: Two special cases, L = 1 and L = 2, are of great interest.For L = 1, f 2 is: and for L = 2, f 2 becomes: Referring to (32), it is easily proved that f 2 < f 1 , and if λ L → 0, then, f 2 → f 1 .Consequently, the phase noise profile of a fractional-N PLL behaves similarly to that of an integer-N counterpart.Assuming B = 0.3 for the type-II PLL with design parameters used in Section III-A, Figure 7 shows plots of S (f ) and its estimation as well as the reference and VCO phase noise profiles and the quantization noise.This plot verifies the accuracy of this approximation for the major part of the plot.It is noteworthy that if the modulus signal is randomized, the mechanism of spur generation in the fractional-N PLL is the same as the integer-N PLL.Therefore, (55)-( 20) can be used to characterize the fractional-N PLL in terms of its output spurs.

IV. THE IMPACT OF PLL NOISE AND SPURS ON OOBE
The subsequent analysis in this section focuses on evaluating the impact of phase noise and spurs introduced by integer-N and fractional-N PLLs on OOBE.Referring back to Figure 1, the PLL produces quadrature LO signals, g I (t) and g Q (t), with identical PSD, S G (f ).Similarly, the two input baseband signals, x I (t) and x Q (t), exhibit an identical PSD, S X (f ).
Due to the switching operation of RF mixers, the baseband signal, x(t), is multiplied by a periodic square wave, containing the odd harmonics of the LO signal, g(t) [31].With higher harmonics falling out of band, only the LO's first harmonic is taken into consideration.Therefore, the TX mixer's output signal is roughly assumed to be y(t) = A V,MX × x(t) × g(t), where A V,MX is the mixer conversion gain.A V,MX has no impact on the relative OOB power levels, as it uniformly increases both the in-band and outof-band PSDs.Therefore, without loss of generality, A V,MX is assumed to be equal to one, hence, y(t) = x(t) × g(t).

Assuming g(t) = A C cos (2π f c t + e(t))
, where e(t) denotes the combination of phase noise and spur disturbance whose PSDs were calculated in the previous sections, the output signal is calculated to be: The output waveform in the time domain can be expressed as: Assuming e(t) 1, (36) can be approximated, as follows As proved in Appendix B, the PSD of the baseband representation of the multiplied signal, x(t) × g(t), denoted by S Y (f ), is derived to be where S Y1 (f ) and S Y2 (f ) are the output PSDs due to the PLL phase noise and spurs, respectively.It is worth noting that S Y2 (f ) also contains the effect of the carrier tone.The factor of two in (38) is attributed to the combination of the mixers' outputs to produce the quadrature signal.Just like before, (38) is normalized by 2A C , as it has no effect on the relative OOBE level.Therefore, the forthcoming equations regarding the relative out-of-band emission levels are considered general and can be used for various types of digital modulation schemes such as PAM-2 M , QPSK, and 4 M QAM.

A. INTEGER-N PLL IMPACT ON OOBE
To accurately calculate the PSD of the mixer's output due to the PLL phase noise for a pulse-shaped random digital stream, ( 5) and ( 13) must be convolved.However, a closed-form expression cannot be obtained.Intuitively, one can surmise that as the roll-off factor decreases, the signal bandwidth will reduce, which leads to an increase in OOBE.More precisely, the input PSD around the edges of the band (i.e., around f = ± 1+β 2T b ) increases, thus, producing a larger output PSD when convolving with the PLL spectrum for |f | > 1+β 2T b .Therefore, the worst-case OOBE occurs when β = 0. To capture the worst-case scenario, the input signal PSD, S X (f ), is approximated by a rectangular spectrum whose PSD is expressed by (7) (cf. Figure 8(a)).Additionally, S E (f ), the spectrum of e(t), is shown in Figure 8(b) in which blue curve and red impulses denote phase noise and spurs, respectively.The PSD of the mixer's output S Y1 (f ) due to PLL phase noise is obtained approximately by convolving ( 7) and ( 16), resulting in where f max and f min are defined as max(f 1 , f BB ) and min(f 1 , f BB ), respectively.Furthermore, S Y2 (f ) is obtained by convolving (7) and (20), and adding the baseband spectrum, S X (f ), to the result, leading to: In practice, for an integer-N PLL, the channel spacing is either the same or an integer multiple of the PLL reference frequency [1].Therefore, f r ≥ 2f BB , simplifying (40) to: Referring to (38), the normalized PSD of the mixer's output, S Y (f ), is obtained by adding (39) and (41).In the special case where the PLL spurs are negligible, the normalized PSD for positive frequencies is approximately calculated by ( 42), shown at the bottom of the next page.It is inferred from (42) that decreasing f 1 will lower OOBE.This, in turn, implies that reducing the VCO phase noise or the PLL loop bandwidth through the reduction of R 1 , K VCO , or I p can improve OOBE.This, however, will decrease ζ , making the PLL loop less stable.Therefore, one way of improving OOBE is to reduce K VCO and I p , while boosting R 1 with the same scaling factor so as to keep ζ constant, and thus, avoid degrading loop stability.This procedure, however, narrows the tuning range of both the VCO and the PLL.It should also be noted that if the PLL loop bandwidth becomes excessively small, ( 16) cannot predict the PLL phase noise profile, accurately.Hence, (42) becomes inaccurate.

B. FRACTIONAL-N PLL IMPACT ON OOBE
Similarly, for the fractional-N PLL, the mixer's output PSD due to PLL phase noise, S Y1 (f ), is derived by replacing every η in (13) with S I (f ) in ( 26), and subsequently, convolving the result with (5).However, the resulting integration does not lead to a closed-form expression, and can only be calculated numerically.Similar to Section IV-A, to obtain an insightful closed-form expression for the spectrum of the mixer's output due to PLL phase noise, ( 7) is convolved with (31), yielding (43), shown at the bottom of the page.
As mentioned earlier, the spur generation mechanism in a fractional-N PLL follows that in an integer-N counterpart.Consequently, (41) is used to express the mixer's output PSD due to PLL's spurs as well as the carrier tone.Therefore, the normalized PSD of the mixer's output, S Y (f ), is obtained by adding ( 41) and (43).For instance, in the case where the PLL spurs are negligible, the normalized PSD for positive frequencies is approximately calculated by ( 44), shown at the bottom of the page.This equation suggests that a reduction in QNF and/or the PLL bandwidth directly lowers the OOBE level.

V. THE EFFECT OF CIRCUIT NONLINEARITY & BANDWIDTH LIMITATION ON OOBE
TX circuit blocks, especially those up the TX chain close to the antenna, exhibit nonlinearity that degrades the TX performance in terms of EVM [14].Additionally, nonlinearity causes spectral regrowth, which elevates the OOBE level.PA, handling large signals, is arguably the most critical TX block in terms of nonlinearity.The PA nonlinearity can be captured by Taylor's series approximation of its nonlinear input-output characteristic [1], as follows: where a k is an empirical coefficient describing the k th -order nonlinearity of the PA, and y(t) and w(t) represent the PA's input and output signals, respectively.Approximating (45) with its first four terms where a 0 is the output's DC voltage, a 1 is the linear voltage gain of the system, and a 2 and a 3 are the second-and third-order nonlinearity coefficients, respectively, leads to [1] The second-order nonlinearity is significantly reduced by adopting fully-differential signaling.Disregarding the DC and second-order terms, the power-series is simplified to Moreover, with the input being a zero-mean random process, the output PSD is related to the input PSD, as follows [15] S Y (f ) where P Y is the total signal power that is fed to the PA, G P = A 2 V is the PA's available power gain, P IIP3 = V 2 IIP3 , and S Y (f ) and S W (f ) are the PSD of input and output signals, respectively.P Y is readily obtained to be P Y = 2×G MX ×P X where G MX is the power gain of the mixer, and the factor of two arises due to combination of two quadrature signals after the mixers.It is noteworthy that (48) is not specific to the PA nonlinearity.In fact, this equation can express the output PSD of an entire system consisting of cascaded blocks if G P and P IIP3 denote the system's power gain and IIP3, respectively.
In a conventional TX chain, the PA is fed by the signal emerging from the output of the power combiner (cf. Figure 1), where the two quadrature upconverted PAM signals are combined.Therefore, S Y (f ), calculated in the preceding section, is the PA input PSD, which can be used in (48) to calculate the PA output PSD.However, the closed-form expression for this case cannot be obtained, necessitating numerical calculations.To address this issue, it is assumed that the PA is fed by the modulated PAM signal which is up-converted to the carrier frequency using a noiseless LO.This assumption enables us to obtain a closed-form expression for the PA output PSD.Additionally, it allows a comparison between the contributions of PLL phase noise and circuit nonlinearity to OOBE.Using (7) as the PA input signal's PSD, the output PSD, based on (48), is derived to be PAs having multiple stages are ubiquitously used in highfrequency (e.g., mm-wave and THz) TX architectures, where each stage has an L-C resonant circuit filtering the output signal.For instance, Figure 9 illustrates a PA incorporating k stages each having a second-order band-pass filter (BPF).Therefore, the overall PA circuit uses a 2k th -order BPF.Assuming each stage of filtering has the same center frequency of f c and quality factor of Q, the normalized magnitude of the transfer function of each filtering stage is derived to be where f c = 1/2π √ L D C D , and L D and C D are the inductor and capacitor values, respectively [16].For frequencies close to f c (i.e., 0.9f c < f < 1.1f c ), (50) is approximated by where f is the offset frequency from the center frequency, f c .Therefore, Taking into account for the PA bandwidth, the baseband representation of the output signal spectrum is: where S W (f ), the PA's output PSD disregarding its limited bandwidth, was calculated in (49).The normalized PSD of the PA output for positive frequencies is obtained in (53), shown at the bottom of the next page.It is observed that linearizing the PA (i.e., increasing P IIP3 ) and increasing the order and quality factor of the filter reduces OOBE, especially at larger offset frequencies.

VI. SIMULATION RESULTS
To examine the accuracy of the proposed analytical study, comprehensive system-level simulations have been performed, and the important results are presented in this section.

A. BASEBAND
A raised-cosine pulse-shaped PAM-4 signal with a baud rate of 100 MS/s was used as baseband digital stream.The rolloff factor was considered to be β = 0.  amplitude of 100 mV. Figure 10(b) displays a comparison between the PSDs of the baseband signal acquired from simulation (in red squares) and closed-form expression in (7) (in blue diamonds), where these results follow one another with 1-dB accuracy within the signal's bandwidth.The out-of-band PSD of the simulated baseband signal is more than 40 dB weaker than the in-band PSD.The discrepancy between analysis and simulation results is attributed to the limited number of simulated symbols (i.e., 800 symbols), which is imposed by the finite storage capacity of our computational infrastructure.Furthermore, the simulated eye diagram of the transmitted baseband signal is shown in Figure 11, where the horizontal and vertical eye-openings for the simulated 800 symbols are around 2 ns and 200 mV, respectively.

B. INTEGER-N FREQUENCY SYNTHESIZER
A behavioral model of a type-II integer-N PLL was constructed in MATLAB.The PLL loop parameters were chosen, as follows:  Consequently, the damping factor, natural frequency, and loop bandwidth of this PLL are calculated to be 1.96, 3.47 MHz, and 13.6 MHz, respectively.The resulting red squares in Figure 13 represent the PLL output phase-noise profile, while, the green circles and blue diamonds depict the exact and approximate expressions calculated in ( 13) and (31), respectively.It is evident that the approximate phase noise expression derived for an integer-N PLL in (31) accurately tracks the actual simulated counterpart.
To up-convert the baseband signal, a mixer driven by the PLL with a conversion gain of 3 dB was employed.Figure 14 illustrates the normalized PSD of the mixer's output versus offset frequency from the carrier.The simulated results are represented by the blue diamonds, whereas the red squares represent the PSD derived from the proposed theoretical framework.Comparing the simulated results with the theoretical expressions demonstrates the accuracy of the theoretical derivations.

C. FRACTIONAL-N FREQUENCY SYNTHESIZER
A behavioral model of a type-II fractional-N PLL was simulated in MATLAB.The PLL loop parameters including the charge pump current and VCO gain were kept the same as those used for the integer-N PLL.Additionally, a modulator was employed to vary the division ratio of the feedback path such that the average division ratio was 280.2, resulting in a 28.02 GHz output signal from a 100 MHz reference clock.The modulator and fractional-N PLL models used in these simulations were obtained from [32], which provided a comprehensive PLL model considering the effects of instantaneous division ratio variations.A secondorder modulator was utilized to generate a random binary sequence that averaged to 0.2, while shaping the phase noise profile.The shaped quantization noise PSD of the modulator, expected to have minimal power at low frequencies, is depicted in Figure 15 using blue diamonds.In the same figure, ( 24) is also plotted using red squares.Additionally, the phase noise of the fractional-N PLL was analyzed in Figure 16.
The baseband signal was up-converted using the previously mentioned mixer.Figure 17 displays the normalized PSD of the mixer's output versus offset frequency from the carrier tone.The blue diamonds represent the simulation results, while the red squares depict the PSD derived from the theory developed throughout this paper.

D. CIRCUIT NONLINEARITY
To evaluate the impact of circuit nonlinearity on OOBE, a behavioral model for the TX was developed.The TX chain has a gain of 20 dB and an IIP3 of 7.5 dBm. Figure 18 depicts the normalized PSD of the TX's output signal at various offset frequencies.In this figure, the simulation results are represented by the blue diamonds, while the red squares depict the PSD derived from (49), which is based on the theoretical framework developed in this study.By comparing Figures 14, 17, and 18, it becomes apparent that the circuit nonlinearity affects OOBE at low offset frequencies.Nevertheless, the PLL phase noise can potentially have a more detrimental impact at higher offset frequencies.

VII. DESIGN GUIDELINES
Based on the simulation results and the theory developed in this paper, several guidelines assisting a TX designer, are extracted.These are outlined, as follows: 1) When a TX incorporates an integer-N frequency synthesizer, reducing VCO phase noise or the PLL loop bandwidth by decreasing R 1 , K VCO , or I p improves OOBE.Nonetheless, these modifications decrease ζ , adversely affecting the loop's stability and settling time.To avoid this, a viable approach involves reducing both K VCO and I p while concurrently increasing R 1 by the same scaling factor to maintain a constant ζ .However, it is noteworthy that this procedure may reduce the PLL locking range.2) When a TX utilizes a fractional-N synthesizer, reducing QNF proves to be pivotal in enhancing OOBE.It is worth noting that in practical scenarios where a modulator is employed to shape quantization noise, QNF decreases as the reference frequency increases.3) Due to the absence of quantization noise, transmitters incorporating integer-N frequency synthesizers typically exhibit lower OOBE.Consequently, integer-N frequency synthesizers present a distinct advantage in terms of minimizing OOBE.4) Linearizing PA and increasing the BPF's order and quality factor improve OOBE.However, designing on-chip high-quality factor BPFs remains extremely challenging.
5) At low and high offset frequencies, OOBE is primarily affected by the PA nonlinearity and PLL phase noise, respectively.Consequently, the application's requirements determine whether the PA linearity and/or the PLL phase noise should be optimized to achieve the desired performance.

VIII. CONCLUSION
This paper presented an analysis of out-of-band emission in TXs incorporating digital modulation schemes caused by three primary impairments: (1) the Integer-N and Fractional-N PLL's phase noise and spurs, (2) circuit nonlinearity, and (3) bandwidth limitation.Closed-form expressions for out-ofband emission at an arbitrary frequency offset were obtained.
The developed theory and the simulation results show that, typically, the integer-N frequency synthesizer outperforms the fractional-N counterpart in terms of OOBE, especially at low offset frequencies.Furthermore, the PA nonlinearity mostly affects OOBE at low offset frequencies, whereas the PLL nonidealities impact OOBE even at very large offset frequencies.The proposed analytical expressions can be used in the link budget calculations of TXs to determine the required phase noise of the PLL and the nonlinearity of different circuit blocks within the TX chain.

FIGURE 1 .
FIGURE 1.A basic direct-conversion TX block diagram.

FIGURE 2 .
FIGURE 2. Representation of a raised-cosine pulse in (a) time, and (b) frequency domain.

FIGURE 3 .
FIGURE 3. PLL phase noise profile and its approximation.

FIGURE 4 .FIGURE 5 .
FIGURE 4. The double-sided PSD of the PLL output signal.
)-6(b) show the variation of QNF with respect to reference clock frequency for three distinct values of B.

FIGURE 6 .FIGURE 7 .
FIGURE 6. QNF variation with respect to reference clock frequency when (a) no pulse shaping is being used, and (b) second-order pulse shaping is utilized.

Figure 10
(a)  shows the simulated baseband PAM-4 signal with a minimum
pF, and N = 280.The reference was a 100 MHz crystal oscillator with a flat phase noise profile at -160 dBc/Hz.The constituent VCO, oscillating at the center frequency of 28 GHz, employed a varactor-based LC cross-coupled pair topology with K VCO = 4GHz/V (c.f.Figure12(a)), and