# Mixer Sequence Design For N-Path Filters

A bandpass filter includes a plurality of parallel paths, each receiving the input signal to the bandpass filter. Each path includes a first mixer, a low-pass filter, and a second mixer. The first mixer in each path is coupled to receive the input signal and mixes the input signal with a periodic mixer sequence having a period that is divided into a plurality of time slots. The mixer value is constant during each time slot. The low-pass filter in each path is operable to filter an output of the associated first mixer. The second mixer in each path is coupled to receive an output of the associated low-pass filter and mixes said filter output with a periodic mixer sequence having a period that is divided into a plurality of time slots, wherein again the mixer value is constant during each time slot. A summer sums the outputs of the second mixers of each of the paths to generate an output of the bandpass filter.

**Description**

**BACKGROUND**

Bandpass filters are used in many applications including radio frequency receiver paths and bandpass delta-sigma analog-to-digital converters. Typical bandpass filter implementations require inductors, and it is difficult to implement high quality, appropriately sized, inductors in complementary metal oxide semiconductor (CMOS) processes. Other options exist for implementing bandpass filters, but they tend to have issues with performance and/or power, or require alternative process technologies.

N-path filters are a practical method for implementing high-Q bandpass filters in modern CMOS processes without inductors using a combination of mixers and low-pass filters. The Q factor is the ratio of the center frequency of the filter to the pass band bandwidth. The basic structure of an N-path filter is multiple paths, each path composed of a mixer, filter and mixer, summed together to form the filter output. With trends in process scaling leading to higher switching frequencies, N-path filters are a viable option for integrated bandpass filter designs with center frequencies of interest in current communication standards. As the center frequency of the filter is decoupled from the bandwidth of the filter, high Q values are achievable. The mixer sequences have a high impact on the performance of the N-path filter. Therefore optimizing practically realizable mixer sequences allows optimization of the N-path filter.

**SUMMARY**

A bandpass filter in accordance with one illustrative embodiment of the present invention includes a plurality of parallel paths, each receiving the input signal to the bandpass filter. Each path includes a first mixer, a low-pass filter, and a second mixer. The first mixer in each path is coupled to receive the input signal and mixes the input signal with a periodic mixer sequence having a period that is divided into a plurality of time slots. The mixer value is constant during each time slot. The low-pass filter in each path is operable to filter an output of the associated first mixer. The second mixer in each path is coupled to receive an output of the associated low-pass filter and mixes said filter output with a periodic mixer sequence having a period that is divided into a plurality of time slots, wherein again the mixer value is constant during each time slot. A summer sums the outputs of the second mixers of each of the paths to generate an output of the bandpass filter.

Another embodiment of the invention is directed to a bandpass filter that includes a first path and a second path. The first path includes a first mixer, a first low-pass filter, and a second mixer. The first mixer mixes an input signal with a sampled cosine signal to produce a first mixed signal. The first low-pass filter is operable to low-pass filter the first mixed signal to produce a first filtered signal. The second mixer mixes the first filtered signal with the sampled cosine signal to produce a second mixed signal. The second path includes a third mixer, a second low-pass filter, and a fourth mixer. The third mixer mixes the input signal with a sampled sine signal to produce a third mixed signal. The second low-pass filter low-pass filters the third mixed signal to produce a second filtered signal. The fourth mixer mixes the second filtered signal with the sampled sine signal to produce a fourth mixed signal. A summer sums the second mixed signal and the fourth mixed signal to produce an output of the bandpass filter.

Another embodiment of the invention is directed to a bandpass filter that has N parallel paths, each path arranged to receive the input signal to the bandpass filter. Each path includes a first mixer, a low-pass filter, and a second mixer. The first mixer in each path is coupled to receive the input signal and mixes the input signal with a periodic two-level mixer sequence having a period of length T that is divided into M time slots, where M is an integer greater than one. The mixer value is constant during each time slot. Assume M is an even number. The number of paths N is greater than or equal to M/2. Only the first M/2 paths are active, where the mixer sequence p of the first mixer of the n^{th }path (1≦n≦M/2) is chosen to satisfy

The low-pass filter in each path is operable to filter an output of the associated first mixer. The second mixer in each path is coupled to receive an output of the associated low-pass filter and mix said filter output with a periodic two-level mixer sequence having a period of length T that is divided into M time slots. The mixer value is constant during each time slot. The mixer sequence q of the second mixer of the n^{th }path (1≦n≦M/2) is chosen to satisfy

A summer sums the outputs of the second mixers of each of the paths to generate an output of the bandpass filter.

**BRIEF DESCRIPTION OF THE DRAWINGS**

**DETAILED DESCRIPTION**

The present invention is directed generally to an N-path bandpass filter having mixer sequences that are constrained to a staircase sequence and to a two-level sequence.

**100**. The first path includes a mixer **105**, a low-pass filter **120**, and a second mixer **135**. The second path includes a mixer **110**, a low-pass filter **125**, and a second mixer **140**. The N^{th }path includes a mixer **115**, a low-pass filter **130**, and a second mixer **145**. The first mixers **105**, **110**, **115** in each path have a mixer signal p^{(n)}(t), and the second mixers **135**, **140**, **145** in each path have a mixer signal q^{(n)}(t). The mixers have a period T which determines the center frequency Ω_{0}=2π/T of the bandpass filter. The products of each path are summed together by summer **150** to form the filter output. The mixers transform the low-pass filter shape to a pass band around Ω_{0}, where the double-sided bandwidth of the low-pass filter is the same as the bandwidth of the bandpass filter. In an illustrative embodiment, the N-path filter **100** is implemented in complementary metal-oxide-semiconductor (CMOS) integrated circuit(s). But other technologies, including other semiconductor technologies, can be used as well.

It can be shown mathematically that the N-path filter **100** transforms a low-pass-filter into a bandpass filter. Defining X(jΩ) as the input and Y(jΩ) as the output spectrum of the N-path filter, and denoting H(jΩ) as the low-pass filter, the output of the N-oath filter can be represented as:

in which the input and mixer related terms are

where {circumflex over (p)}_{m}^{(n) }and {circumflex over (q)}_{m}^{(n) }are the m^{th }Fourier series coefficients of p^{n}(t) and q^{n}(t), respectively. Typically, the bandwidth BW<<Ω_{0 }and Eq. (1) implies that the output only has significant power in frequencies ±BW/2 around the harmonics of Ω_{0}.

For simplicity, it can be assumed that the power is flat in frequencies ±BW/2 around a harmonic, so only the midpoint of each band (i.e., Y(j·lΩ_{0})) is considered. When Ω=lΩ_{0 }in Eq. (1), only the r=l term remains. Since H(j0) is the same scale factor for all harmonics, let H(j0)=1 to obtain

α(m, l) can be viewed as the transfer coefficient from the m^{th }harmonic in the input to the l^{th }harmonic in the output.

It can further be assumed for simplicity's sake that the stationary input signals at different harmonics are uncorrelated and both X(j·mΩ_{0}) and X^{2}(j·mΩ_{0}) have zero mean. Denoting E{·} as the average operator, the output signal power spectral density (PSD) at the l^{th }harmonic is:

The in-band output corresponds to l=1 and has two components: the in-band signal and the folded harmonic. The desired in-band signal, which is the output of a traditional bandpass filter, corresponds to the term l=m=1 and has average power

*P*_{signal}*=E{|X*(*jΩ*_{0})|^{2}}·|α(1, 1)|^{2}· (6)

The unwanted folded harmonics can be viewed as interference to the in-band signal. They correspond to terms with l=1, m≠1 in (4) and have average total power

In addition to the in-band output, the N-path filter typically has out-of-pass-band outputs around the harmonics of Ω_{0}. For the l^{th }harmonic (l≠±1), the average out-of-band power is

The above analysis shows that the N-path filter transforms a low-pass filter to a bandpass filter with two nonidealities: in-band harmonic folding and out-of-band signal residue. **200** that reduces these two nonideal effects. The bandpass filter **200** employs a loose pre-low-pass filter **210** and a loose post-low-pass filter **230** around the N-path filter **220**, such as the N-path filter **100** of _{0}, the pre-low-pass filter **210** attenuates signals at high harmonics to avoid folding onto the in-band signal and the post-low-pass filter **230** removes residual out-of-band signal power.

Alternatively, as α(m, l) in Eqs. (6-8) depends on the Fourier coefficients of the mixer signals, it's possible to design mixer signals that reduce the in-band harmonic folding and the out-of-band signal residue such that the requirements on the pre and post low-pass filters **210** and **230** are reduced or eliminated. The design of the mixer sequences p^{(n)}(t) of the mixers **105**, **110**, **115**, and the mixer sequences q^{(n)}(t) of the mixers **135**, **140**, **145** for this purpose is explored below.

In an illustrative embodiment of the present invention, the mixer sequences p^{(n)}(t) of the mixers **105**, **110**, **115**, and the mixer sequences q^{(n)}(t) of the mixers **135**, **140**, **145** have periodic staircase sequences. Each period is split into M equal time slots and each mixer value is constant within each time slot. **300**, an input signal, such as signal x(t) shown in **310**, the input signal is provided to each of a plurality of parallel paths. Each path includes a first mixer, a low-pass filter, and a second mixer. At block **320**, in each path, the first mixer **105**, **100**, **115** mixes the input signal with a periodic mixer sequence having a period that is divided into a plurality of time slots. The mixer value is constant during each time slot. At block **330**, the low-pass filter **120**, **125**, **130** in each path low-pass filters the output of the associated first mixer **105**, **110**, **115**. At block **340**, the second mixer **135**, **140**, **145** in each path receives the output of the associated low-pass filter and mixes said filter output with a periodic mixer sequence having a period that is divided into a plurality of time slots, wherein the mixer value is constant during each time slot. At block **350**, the summer **150** sums the outputs of the second mixers **135**, **140**, **145** of each of the paths to generate an output of the bandpass filter.

For purposes of the present invention, two criteria are considered in evaluating N-path filters: in-band signal-to-noise ratio (SNR) and out-of-band harmonic power ratio, if the folded harmonics are considered as in-band noise, then the goal is to maximize the in-band SNR=P_{signal}/P_{folded}. Regarding the out-of-band harmonic power ratio, for l≠±1, the goal is to minimize R_{out}(l)=P_{out}(l)/P_{signal}.

The following lemma and corollary, which put an upper limit on the achievable in-band SNR and a lower limit on the out-of-band harmonic power ratio, are used in design of the mixer sequences p^{(n)}(t) of the mixers **105**, **110**, **115**, and the mixer sequences q^{(n)}(t) of the mixers **135**, **140**, **145**. The lemma states that, for M-slot staircase mixer sequences, the harmonic power ratio for the lth harmonic is lower bounded by

and the lower bound is achieved if

α(*m, l*)=0, for 0*≦m<M, *0*≦l<M, *

except for (*m, l*)=(1, 1 ) or (*M−*1*, M−*1). (10)

The ratio α(bM+1, cM+1)/a(1, 1) is independent of the input, where a(1, 1) controls P_{signal }and α(bM+1, cM+1) controls the power contributed from the (bM+1)^{th }input harmonic to P_{out }(cM+1). The bound is achieved when no other input harmonics contribute to the (cM+1)^{th }output harmonic. The case l=cM−1 is due to spectrum symmetry. Setting l−1 in the lemma leads to the corollary: with M-slot staircase mixer sequences, the in-band signal-to-noise ratio is upper bounded by

and the upper bound is achieved if

α(*m, *1)=0, for *m≠*1, 0*≦m<M. * (12)

**Staircase Sequences**

In one illustrative embodiment of the M-slot staircase mixer sequences of the present invention, the signal amplitude can vary from time slot to time slot and from path to path, and there is no constraint on the relationship between the mixer signals on different paths or the number of paths N. In this scenario, a two-path sampled quadrature filter achieves both the optimal in band signal-to-noise ratio and the optimal harmonic power ratio at each harmonic frequency, among all N-path filters with M-slot staircase mixer sequences. In the period [0, T], the mixer sequences in the two-path sampled quadrature filter have values

*p*^{(1)}(*t*)=*q*^{(1)}(*t*)=cos(2 *πm/M*)

*p*^{(2)}(*t*)=*q*^{(2)}(*t*)=sin(2 *πm/M*) (13)

where (mT/M)≦t<((m+1)T/M) and 0≦m≦M−1.

Thus a bandpass filter according to this embodiment comprises just two paths. Referring again to **105**, low-pass filter **120**, and second mixer **135**, and a second path comprising first mixer **110**, low-pass filter **125**, and second mixer **140**. Both mixers **105** and **135** in the first path utilize a mixer sequence comprising a sampled cosine signal. Both mixers **110** and **140** in the second path utilize a mixer sequence comprising a sampled sine signal.

To prove that this two-path scheme employing sampled sine and cosine mixer sequences is the optimal N-path filter that uses periodic staircase mixer sequences, it is necessary only to test whether the sequences of (13) satisfy (10) and (12). Note that for 0≦m≦M−1, the only nonzero terms of Fourier series coefficients of p^{(n)}(t) and q^{(n)}(t) in (13) are {circumflex over (p)}_{1}^{(n)}, {circumflex over (p)}_{M−1}^{(n)}, {circumflex over (q)}_{1}^{(n)}, and {circumflex over (q)}_{M−1}^{(n)}. Thus, for 0≦m, l≦M−1, the only possible nonzero terms of α(m, l) are α(l, **1**), α(l, M−1), α(M−1, 1) and α(M−1, M−1). Direct calculation can verify that α(l, M−1)=α(M−1, 1)=0. As such, (10) is satisfied and the minimum harmonic power ratio at each harmonic is achieved.

Constraint (12) for the optimal in-band signal-to-noise ratio is a special case of (10) with l=1 which is already satisfied. Thus, the sampled sine and cosine mixer signals of (13) achieve both maximum in-band signal-to-noise ratio and minimum harmonic power ratio at each harmonic.

**400**, an input signal, such as signal x(t) shown in **410**, **420** and **430**, includes a first mixer, a low-pass filter, and a second mixer. At block **410**, the first mixer mixes the input signal with a sampled cosine signal to produce a mixed signal. At block **420**, the low-pass filter low-pass filters the mixed signal to produce a low-pass filtered signal. At block **430**, the second mixer mixes the low-pass filtered signal with the sampled cosine signal to produce a second mixed signal. The second path, represented by blocks **440**, **450** and **460**, also includes a mixer, a low-pass filter, and a second mixer. At block **440**, the second path's first mixer mixes the input signal with a sampled sine signal to produce a mixed signal. At block **450**, the low-pass filter filters the mixed signal to produce a low-pass filtered signal. At block **460**, the second mixer of the second path mixes the low-pass filtered signal with the sampled sine signal to produce a second mixed signal. At block **470**, a summer sums the mixed signal received from the first and second paths to produce an output of the bandpass filter.

It is noted that, if the mixers of the N-path filter such as that shown in

**Two-Level Sequences**

In another illustrative embodiment of the M-slot staircase mixer sequences of the present invention, the mixer sequences are further constrained to taking on one of only two values in each of the M slots. Thus:

p^{(n)}(t) ∈ {1, −1}, q^{(n)}(t) ∈ {A_{n}, −A_{n}}, (14)

where A_{n }is a constant gain for the n^{th }path. Limiting the mixers to two levels makes them easier to implement in analog.

In the following design, it is assumed for the sake of simplicity of explanation that M is an even number. If M is odd, the results are similar as will be explicitly described subsequently. For an even M, the cases of N≧M/2 and N<M/2 paths are separately considered. Additionally, only the in-band signal-to-noise ratio criterion is considered in the analysis of the two-level sequences.

Two-Level Sequences with N≧M/2 Paths

Among N-path filters whose mixer sequences have M slots per period and are constrained to taking on one of only two values in each of the M slots, the M/2-path filter with the following class of mixer sequences achieves the optimal in-band signal-to-noise ratio:

*p*^{(n)}(*t*)=*p*^{(1)}(*t*−((*n−*1)*T/M*)), (15)

*g*^{(n)}(*t*)=*q*^{(1) }(*t*−((*n−*1)*T/M*)), (16)

*p*^{(1)}(*t***30** (*T/*2))=−*p*^{(1)}(*t*). (17)

The antisymmetric condition of Eq. (17) indicates that p^{(n)}(t) has no even harmonics. Thus {circumflex over (p)}_{2m}^{(n)}=0 and α(2m, 1)=0. The delay relationships in Eqs. (15) and (16) result in a phase factor in the Fourier series coefficients and it can be verified that α(2m+1, 1)=0 for 1≦m<M/2. Thus, (12) is satisfied and the optimal in-band SNR is achieved.

In particular, the half-plus half-minus (HPHM) sequences

satisfy the constraints of (15-17) and have optimal in-band SNR. These sequences are particularly implementation friendly as there are only two level changes in one period in each path, and mixer sequences in consecutive paths have a delay of one slot.

**500**, an input signal, such as signal x(t) shown in **510**, the input signal is provided to each of a plurality of parallel paths. Each path includes a first mixer, a low-pass filter, and a second mixer. At block **520**, in each path, the first mixer **105**, **100**, **115** mixes the input signal with a periodic two-level mixer sequence having a period that is divided into M time slots, wherein N≧M/2. The mixer value is constant during each time slot and the mixer sequence p satisfies Eq. (15) and Eq. (17). At block **530**, the low-pass filter **120**, **125**, **130** in each path low-pass filters the output of the associated first mixer **105**, **110**, **115**. At block **540**, the second mixer **135**, **140**, **145** in each path receives the output of the associated Sow-pass filter and mixes said filter output with a periodic two-level mixer sequence having a period that is divided into M time slots, wherein the mixer value is constant during each time slot and wherein N≧M/2. The mixer sequence q satisfies Eq. (16). At block **550**, the summer **150** sums the outputs of the second mixers **135**, **140**, **145** of each of the paths to generate an output of the bandpass filter.

It is noted that additional paths beyond M/2 do not provide a gain for in-band signal-to noise ratio. However, out-of-band signal rejection could potentially be improved. Additionally, the optimal sequences are independent of the input signal power spectral density.

Two-Level Sequences with N<M/2 Paths

N-path filters whose M-slot mixer sequences are constrained to taking on one of only two values in each of the M slots, and where the number of paths N is restricted to N<M/2, are now considered. Only the in-band signal-to-noise ratio criterion is considered in the analysis of these two-level sequences. In contrast to the previous results, an input signal independent mixer sequence is not obtained under these constraints. Instead, a heuristic optimization algorithm is proposed.

Let v_{k}^{(n) }and w_{k}^{(n) }represent the values in the kth time slot (1≦k≦M) of the mixer sequences p^{(n)}(t) and q^{(n)}(t) respectively. The in band signal-to-noise ratio has the form of

Since the Fourier coefficients {circumflex over (p)}_{m}^{(n) }and {circumflex over (q)}_{1}^{(n) }are linear in the values of v_{k}^{(n) }and w_{k}^{(n)}, respectively, α(m, 1) is a bilinear form with respect to v_{k}^{(n) }and w_{k}^{(n)}. Therefore, the powers of the signal and folded harmonic are both quadratic forms with respect to either v_{k}^{(n) }and w_{k}^{(n)}. The total order of 4 is a challenge for optimizing the signal-to-noise ratio per Eq. (21).

To reduce the order of the objective function, an iterative two-part heuristic algorithm is used. The first part (i.e., part 1) optimizes over v_{k}^{(n) }with w_{k}^{(n) }held constant. The second part (i.e., part 2) optimizes over w_{k}^{(n) }with v_{k}^{(n) }held constant.

The optimization problem of part 1 can be written as

where S and N are positive semidefinite matrices dependent on N, M, w_{k}^{(n) }and the input signal power spectral density E{|X(j·mΩ_{0})|^{2}}. If Nv≠0 for all “binary” v vectors, it can be shown that the optimal objective function of Eq. (22) has the value of λ if and only if the following problem

has a maximum of 0. Problem (23) is an unconstrained binary quadratic programming problem and can be approximately solved by greedy local search. The solution of (23) never decreases signal-to-noise ratio, which typically leads to convergence. Thus, the optimization problem in part 1 can be heuristically solved by an iterative algorithm.

**600**, the vectors v and w are initialized with the half-plus half-minus sequences of (18)-(20). At block **610**, the current λ is computed as λ=(v^{T}Sv)/(v^{T}Nv). At block **620**, problem (23) is solved with the current λ. At block **630**, the vector v is updated with the solution to problem (23). At decision block **640**, it is determined if the optimal cost is zero. If the optimal cost is zero, part 1 of the heuristic algorithm terminates, as shown at block **660**. If the optimal cost is not zero, it is determined whether the limit on iteration steps has been reached at decision block **650**. If the limit on iteration steps is reached, then part 1 of the heuristic algorithm terminates at block **660**. If the limit on iteration steps is not reached, the algorithm is repeated starting at block **610**.

For the second part of the heuristic algorithm, since the in-band signal-to-noise ratio involves only {circumflex over (q)}_{1}^{(n) }the optimization is performed on [{circumflex over (q)}_{1}^{(1)}, . . . , {circumflex over (q)}_{1}^{(N)}]∈^{N}. Equation (21) is the ratio of semidefinite quadratic forms with respect to {circumflex over (q)}_{1}^{(n)}, hence, its solution is available in closed form. After solving for the optimal {circumflex over (q)}_{1}^{(n)}, its phase is quantized into delays which are multiples of T/M.

The two parts of the algorithm may take multiple iterations to determine a heuristic-based optimal solution for the two-level mixer sequences in each path. There are potentially local maximums and no guarantees of global optimality are provided. In contrast to the staircase mixer sequence or two-level sequence with N≧M/2 paths, the optimal sequence in the case N<M/2 paths depends on the input signal and the number of paths.

In the above analysis of the two-level sequences, the number of time slots per period M is assumed an even number. If M is odd, the derivation is very similar and thus omitted. However, there are differences in certain conclusions, which are stated as follows. For an odd M, the cases of N≧M and N<M paths are separately considered. If N≧M, then the M-path filter with sequences satisfying Eq. (15) and (16) achieves the maximum in-band signal-to-noise ratio among all N-path filters with M-slot two-level mixer sequences. This optimum is independent of the input signal power spectral density; furthermore, additional paths beyond M does not improve in-band signal-to-noise ratio. In contrast, if N<M, the optimal sequences can be obtained by the heuristic optimization algorithm that is introduced in paragraphs [0037]-[0043]. In this case, the optimal sequences depend on the input signal and the number of paths.

Having thus described circuits and methods for implementing an N-path bandpass filter by reference to certain of its preferred embodiments, it is noted that the embodiments disclosed are illustrative rather than limiting in nature and that a wide range of variations, modifications, changes, and substitutions are contemplated in the foregoing disclosure. For example, in an illustrative embodiment of the invention, the N-path filter **100** is implemented in complementary metal-oxide-semiconductor (CMOS) integrated circuit(s), but other technologies, including other semiconductor technologies, can be used as well. Furthermore, in some instances, some features of the present invention may be employed without a corresponding use of the other features. Accordingly, it is appropriate that the appended claims be construed broadly and in a manner consistent with the broad inventive concepts disclosed herein.

## Claims

1-13. (canceled)

14. A bandpass filter comprising: a summer operable to sum the outputs of the second mixers of each of the paths to generate an output of the bandpass filter;

- a plurality of parallel paths, each path arranged to receive an input signal to the bandpass filter, each path comprising: a first mixer coupled to receive the input signal and operable to mix the input signal with a periodic two-level mixer sequence having a period that is divided into a plurality of time slots, wherein the mixer value is constant during each time slot; a low-pass filter operable to filter an output of the associated first mixer; and a second mixer coupled to receive an output of the associated low-pass filter and operable to mix said filter output with a periodic two-level mixer sequence having a period that is divided into a plurality of time slots wherein the mixer value is constant during each time slot; and

- wherein the mixer sequences of the first and second mixers of each path have a period that is divided into M time slots, where M is an integer greater than one, and wherein the number of paths N≧M/2.

15. The bandpass filter of claim 14 wherein the mixer sequence p of the first mixer of the nth path (1≦n≦M/2) satisfies p ( n ) ( t ) = p ( 1 ) ( t - ( n - 1 ) T M ), wherein the mixer sequence q of the second mixer of the nth path (1≦n≦M/2) satisfies q ( n ) ( t ) = q ( 1 ) ( t - ( n - 1 ) T M ), and wherein p ( 1 ) ( t + T 2 ) = - p ( 1 ) ( t ).

16. The bandpass filter of claim 15 wherein p ( 1 ) ( t ) = { 1, 0 ≤ t < ( T / 2 ), - 1, ( T / 2 ) ≤ t < T, and p ( n ) ( t ) = p ( 1 ) ( t - ( ( n - 1 ) T / M ) ), and wherein where 1≦n≦M/2.

- q(n)(t)=p(n)(t),

17. A bandpass filter comprises: a summer operable to sum the outputs of the second mixers of each of the paths to generate an output of the bandpass filter;

- a plurality of parallel paths, each path arranged to receive an input signal to the bandpass filter, each path comprising: a first mixer coupled to receive the input signal and operable to mix the input signal with a periodic two-level mixer sequence having a period that is divided into a plurality of time slots, wherein the mixer value is constant during each time slot; a low-pass filter operable to filter an output of the associated first mixer; and a second mixer coupled to receive an output of the associated low-pass filter and operable to mix said filter output with a periodic two-level mixer sequence having a period that is divided into a plurality of time slots wherein the mixer value is constant during each time slot; and

- wherein the mixer sequences of the first and second mixers of each path have a period that is divided into M time slots, where M is an integer greater than one, and wherein the number of paths N<M/2.

18. The bandpass filter of claim 17 wherein the mixer sequences of the first and second mixers of each path are designed by an iterative optimization method.

19. The bandpass filter of claim 18 wherein said iterative optimization method comprises for each path alternating between optimizing the mixer sequence of the first mixer with the mixer sequence of the second mixer held constant, and optimizing the mixer sequence of the second mixer with the mixer sequence of the first mixer held constant.

20. The bandpass filter of claim 14 wherein the bandpass filter comprises a complementary metal-oxide semiconductor (CMOS) integrated circuit.

**Patent History**

**Publication number**: 20160043756

**Type:**Application

**Filed**: Aug 6, 2014

**Publication Date**: Feb 11, 2016

**Patent Grant number**: 9294138

**Inventors**: Guolong Su (Cambridge, MA), Arthur John Redfern (Plano, TX)

**Application Number**: 14/452,777

**Classifications**

**International Classification**: H04B 1/10 (20060101); H04W 72/04 (20060101); H04B 17/336 (20060101);