Frequency-Domain Analysis of  $N$-Path Filters Using Conversion Matrices

N-path filters are finding increased prominence in recent architectures for tunable transceivers. The clock-programmable center frequency together with programmable baseband bandwidth makes it a natural fit for wide programmability required in software-defined radios and cognitive radios. However, the analysis of such filters remains difficult due to their linear periodically time-varying (LPTV) nature. This brief presents an analysis of N-path filters using conversion matrices. Conversion matrices allow for the analysis of an LPTV circuit with equivalent frequency-domain circuits that can, in turn, be analyzed similar to a linear time-invariant circuit. On applying this method to N-path filters, results already established in prior art are accurately reproduced. Furthermore, the effects of a few important nonidealities, such as clock overlaps, nonideal clock duty cycles, and parasitic elements, are also calculated and verified.

Several techniques to analyze LPTV circuits have been reported.Since the circuit is linear, we can still use the impulse response to characterize it.However, due to the time-varying nature of the circuit, the impulse response is also time varying and, hence, depends on both the time t and the delay τ and is represented by h(t, τ ).Hence, the output, y(t), to the input, u(t), by the convolution, i.e., Since the time variation is periodic, the impulse response h(t, τ ) is periodic in t by the fundamental period of the LPTV system, i.e., T p = 2π/ω p .Hence, it can be expanded as a Fourier series as h(t, τ ) = ∞ n=−∞ h n (τ )e −jnω p t .Substituting for h(t, τ ) in (1), then taking the Fourier transform (in t) and simplifying gives where U (ω) and Y (ω) are the Fourier transforms of the input and output, respectively, H n (ω) is the Fourier transform of h n (t), and the set of transfer functions, i.e., H n (ω) = H n (ω − nω p ), is known as the "harmonic transfer functions" (HTFs).Note that the HTFs are just recentered H n (ω) (by convention).Now, finding the HTFs becomes the goal of the circuit analysis [4]- [12].Analogously, in LTI systems, only a single function, i.e., H 0 (ω), which is the transfer function, is required.Several works derive the HTFs by applying circuit laws in the time domain and then applying a Fourier transform [6], [7].Statespace-based approaches, which treat the circuit as periodically moving through a set of states, have also been shown to be useful [8]- [11].Ultimately, all such prior analysis techniques tend to be ad hoc, and a more systematic approach is desired.This brief highlights an alternative method of analysis that uses the theory of conversion matrices.Conversion matrices have been extensively studied in the literature in the area of computer simulation of complex time-varying circuits [13], and its use has even been extended to nonlinear circuits [14], [15].They have also been proven useful in studying noise in large RF circuits [16].This brief provides a brief overview of conversion matrices.They are then applied to the analysis of a differential N -path filter to highlight their efficacy.

II. CONVERSION MATRICES
To start, let us assume that U (ω) and Y (ω) in ( 2) are bandlimited to ω (−(K +(1/2))ω p , (K +(1/2))ω p ], where K →∞ is a large positive integer.Then, let us define the frequency vector of a frequency transform of a signal x(t) in an LPTV system as the vector where X(ω) is the Fourier transform of x(t), and T p = 2π/ω p is the fundamental period of the system.Then, defining frequency vectors U (ω) and Y (ω) for the input and output of the system, respectively, using (2), it can be shown that Y (ω) = H(ω)U (ω), where the matrix, i.e., is referred to as the conversion matrix (or the "harmonic transfer matrix" [12]) (of the LPTV system) and whose elements are given by H i,j (ω) = H i−j (ω + iω p ), i.e., just frequency-shifted HTFs.By itself, (3) is just the matrix form of (2) and does not give any extra information.However, for simple switching components, the HTFs, and hence H(ω), can be easily derived [13].For example, consider an LPTV capacitor, whose capacitance is varying as C(t) = C(t + T p ).The voltage and the current across the capacitor are related in the time and frequency domains by the following relations: where * is the convolution operator.Since C(t) is periodic, it can be expanded as a Fourier series, and its Fourier transform, i.e., C(ω), can be shown to be a sum of impulses at ω = mω p with amplitudes C m , where m is an integer, and C m is the coefficient of e jmω p t in the Fourier series expansion of C(t): Then, (4) gives Defining vectors V (ω) and I(ω) as before, ( 5) can be written in matrix form as The relation thus obtained, i.e., I(ω) = jCΩ(ω)V (ω), is remarkably similar to the Fourier domain relation for a constant capacitor, i.e., I(ω) = jCωV (ω).Hence, Y C (ω) = jCΩ(ω) can be called the conversion matrix of the "LPTV admittance" of the switching capacitor.This relation is invertible for wellbehaved capacitor variations, i.e., the conversion matrix of the "LPTV impedance" can be obtained by inverting the matrix of the LPTV admittance, and vice versa.Similar relations can be obtained for all other LPTV elements, a subset of which is reported in Table I.Note that all matrices (such as R, L, A, etc.) have the same form as C in (6).
The most important property of the frequency vectors is that they follow linearity.Hence, linear Kirchhoff's law relations hold for time-domain voltage, and current signals hold for their corresponding frequency vectors.This implies that other wellknown LTI results, such as the series and parallel combination of impedances, Thevenin and Norton equivalent circuits, etc., have their conversion matrix equivalents as well.Hence, by using such circuit laws and basic conversion matrices (such as those in Table I), arbitrary LPTV circuits can be easily analyzed to derive their conversion matrices.To illustrate, this technique is applied to N -path filters.

III. N -PATH FILTERS
N -path filters were first proposed in [3].The concept is simple: If a high-frequency RF input signal is first downconverted to around dc, passed through a low-pass filter, and then upconverted back to RF, then the input effectively sees a bandpass filter response.A simple implementation of this concept, the differential bandpass N -path filter is shown in Fig. 1(a) [11] (the differential filter is preferred over the single-ended version as it rejects dc and even harmonics of the switching frequency).The differential RF voltage input is connected though source resistors to a set of N parallel switched capacitor loads in succession, with the switches controlled by nonoverlapping clocks with an ideal duty cycle of 1/N , and a period, i.e., T p , as shown in the figure.The number of paths, i.e., N , is even, and two paths are active at any one instant of time via the closing of two switches so that both ends of the differential input are connected to a load capacitor.The load capacitors, i.e., C, are chosen such that the bandwidth 2/N R s C (the effective resistor seen by each capacitor is N R s /2 due to the duty cycling) is much smaller than ω p .The switches essentially downconvert and upconvert signals, whereas the resistor-capacitor combination behaves similar to a low-pass filter.The differential output voltage is then measured between nodes V o+ (t) and V o− (t) (the common nodes for the switches) and exhibits a bandpass response.This circuit can be easily analyzed using conversion matrices.
Let G i denote the conversion matrix of the LPTV conductance of the switch in the ith branch, G s = R −1 s I represent the source conductance, and Y L = jCΩ(ω) denote the LPTV capacitor admittance.The resultant Fourier domain equivalent circuit is shown in Fig. 1(b), wherein the relevant node voltages and branch currents are labeled.By applying KCL at the node in the ith branch, its voltage, i.e., V i , is given by where j = (i + (N/2)) mod N .Similarly, applying KCL at the positive terminal of the differential output, i.e., V o+ , gives The voltage at the negative terminal of the output, i.e., V o− , can be obtained in exactly the same manner.Hence, the total output differential voltage, i.e., V o = V o+ − V o− , is given by Substituting the value of V i (ω) − V j (ω) [with expressions for V i and V j obtained using (7)] and simplifying, the final output is given by Thus, the LPTV transfer function, i.e., H(ω), that is defined as give the required HTFs {H n (ω)}, as shown in (3).These expressions can be evaluated given numerical circuit components, as illustrated below for an example filter.Note that these expressions are completely general and do not put any restrictions, such as ideal or nonoverlapping clocks.Considering clock nonidealities is very easy in the proposed technique: Only the LPTV conductance matrices, i.e., G i , need to be altered according to the actual resistance variation.To illustrate, consider the case of duty-cycle variation in the clocks.In the ideal case, all the clocks have a duty cycle of 1/N , but suppose the actual duty cycle of the clocks is 1/N (1 − β) (β = 0 being the ideal case).Note that a negative β implies that the clocks are overlapping.Assuming an OFF conductance, i.e., G off , and an ON conductance, i.e., G on , G i can be found from the corresponding periodic conductance variation, i.e., G i (t) [shown in Fig. 1(b)].The coefficient of e jmω p t in the Fourier series of G i (t) is with In the following, let G on /G off = 10 5 and switching frequency ω p = 2π/T p be such that ω p /ω RC = 31.4(similar to [11] for direct comparison), where ω RC = 2/N R s C. A value of K = 2000 was used in (3) for constructing all conversion matrices.All simulation results are from Cadence SPECTRE PSS-PAC simulations, where the switches were modeled as periodically varying conductors using Verilog-A (unless specified).

A. Frequency Response
The HTFs, i.e., {H n (ω)}, give the frequency response of the circuit and can be easily obtained from the entries of H(ω).Thus, the magnitudes of the HTFs of a differential fourpath filter with ideal clock edges and extremely small switch resistance (G on R s /2 = 10 3 ) were calculated and are shown in Fig. 2.These compare very well with results from prior art [11].
These responses (except H o (ω)) can be regarded as folding responses of the filter, whereas H 0 (ω) represents the filter  response.This occurs as the output is essentially "sampled" with a sampling period of T p /N .Hence, inputs at frequencies kN ω p + ω (k is an integer) fold on top of each other due to aliasing.Thus, to reduce this folding, N has to be increased (ideally to infinity).However, this comes at the cost of reduced filtering of the harmonics, i.e., harmonic rejection, in H 0 (ω).To confirm, the calculated magnitude of H 0 (ω) is plotted in Fig. 3 for N = 4 and 8.The values are again in excellent agreement with prior art [11].

B. Impact of Nonidealities 1) Switch ON Resistance:
It is well known that the switch ON resistance, i.e., 1/G on , limits filter attenuation [11].This can be simply understood as follows: Far from the filter center frequency, the capacitors have zero impedance.Hence, the maximum attenuation is approximately −20 . The calculated filter response, i.e., H 0 (ω), plotted in Fig. 4 agrees with this intuition and matches the simulation results.
2) Nonideal Duty-Cycle Clocks: The effect of nonideal duty-cycle clocks (which may even introduce clock overlaps) on H 0 (ω) can be easily studied by using nonzero values of β.Fig. 5 shows the calculated H 0 (ω) for two values of β, alongside simulation results from verifying them.For β = 0.2, i.e., the case where all the switches are OFF for a certain period of time, the filtering response still retains a few peaks, but filter attenuation reduces to about 15 dB.In the case of β = −0.2,i.e., overlapping clocks, the filtering response has completely   degraded.This is a well-known empirical result and occurs due to charge sharing between the load capacitors, but was not predicted by prior art.
It is also interesting to see how the extent of degradation in H 0 (ω) in case of overlapping clocks actually depends on the switch ON resistance.Fig. 6 shows the calculated and simulated H 0 (ω) for various values of switch ON conductance, i.e., G on .It can be readily noted that the degradation is worse for higher values of G on .This is intuitive as higher switch ON resistance reduces the extent of charge sharing between load capacitors.Of course, this comes at the expense of lower filter attenuation due to lower G on , as shown in Fig. 4. Again, this was not predicted in prior art and was only known from simulations [11].3) Switch Parasitic Capacitance: We can also consider parasitic elements.For example, if the switches introduce parasitic capacitors C p at nodes V o+ and V o− in the circuit, the final output can be easily shown to be (11) where Y p = jC p Ω(ω).Note that ( 11) is exactly the same as (9) with the exception of the presence of Y p .If we denote C p = αC, then the calculated and simulated filter responses are shown in Fig. 7. Notice that the filter gain and center frequency has degraded when C p is just 10% of the load capacitance, i.e., C.This is an important result that is yet to be mentioned in prior art (and in fact limits the sizes of switches that can be used, and hence G on ).

C. Real Switches
Finally, the switches were also implemented (in schematic) using a TSMC 65-nm complementary metal-oxidesemiconductor process as transmission gates consisting of a p-type metal-oxide-semiconductor and an n-type metal-oxidesemiconductor transistor with w/L = 0.5 μm/60 nm, and M fingers each, biased at V dd /2, and driving C = 50 pF.They are driven by nonoverlapping clocks operating at 500 MHz with linear rising/falling edges lasting 50 ps each and swinging from 0 to V dd = 1.2 V.For calculations, the switch parasitics and conductance variations were obtained from simulations (their Fourier coefficients were calculated using fast Fourier transforms).The simulated and calculated results for various values of M are shown in Fig. 8 and clearly show the tradeoff between filter attenuation and gain at center frequency due to G on and C p , respectively.

IV. CONCLUSION
In this brief, an analysis of N -path filters using conversion matrices has been presented.The approach uses conversion matrices that allow LPTV circuits to be analyzed using familiar circuit theorems such as Kirchhoff's laws, in a manner similar to LTI circuits.Important results found in prior art were reproduced to verify the analysis technique, and the effects of a few but important circuit nonidealities not considered in prior art were also calculated.

Fig.
Fig. Magnitude of H 0 (ω) for a four-path filter in the case of imperfect dutycycle clocks.

TABLE I BASIC
CONVERSION MATRICES