40,008 research outputs found
A New Technique for the Design of Multi-Phase Voltage Controlled Oscillators
© 2017 World Scientific Publishing Company.In this work, a novel circuit structure for second-harmonic multi-phase voltage controlled oscillator (MVCO) is presented. The proposed MVCO is composed of (Formula presented.) ((Formula presented.) being an integer number and (Formula presented.)2) identical inductor–capacitor ((Formula presented.)) tank VCOs. In theory, this MVCO can provide 2(Formula presented.) different phase sinusoidal signals. A six-phase VCO based on the proposed structure is designed in a TSMC 0.18(Formula presented.)um CMOS process. Simulation results show that at the supply voltage of 0.8(Formula presented.)V, the total power consumption of the six-phase VCO circuit is about 1(Formula presented.)mW, the oscillation frequency is tunable from 2.3(Formula presented.)GHz to 2.5(Formula presented.)GHz when the control voltage varies from 0(Formula presented.)V to 0.8(Formula presented.)V, and the phase noise is lower than (Formula presented.)128(Formula presented.)dBc/Hz at 1(Formula presented.)MHz offset frequency. The proposed MVCO has lower phase noise, lower power consumption and more outputs than other related works in the literature.Peer reviewedFinal Accepted Versio
Identities concerning Bernoulli and Euler polynomials
We establish two general identities for Bernoulli and Euler polynomials,
which are of a new type and have many consequences. The most striking result in
this paper is as follows: If is a positive integer, and
, then we have where
This symmetric relation implies the curious identities of Miki and Matiyasevich
as well as some new identities for Bernoulli polynomials such as
\sum_{k=0}^n\binom{n}{k}^2B_k(x)B_{n-k}(x)=2\sum^n\Sb k=0
k\not=n-1\endSb\binom{n}{k}\binom{n+k-1}{k}B_k(x)B_{n-k}.Comment: 21 page
Consecutive primes and Legendre symbols
Let be any positive integer and let . We
show that for some constanst there are infinitely many integers
with such that
for all ,
where denotes the -th prime, and denotes the
Legendre symbol for any odd prime . We also prove that under the Generalized
Riemann Hypothesis there are infinitely many positive integers such that
is a primitive root modulo for any distinct and
among .Comment: 12 pages, final published versio
A combinatorial identity with application to Catalan numbers
By a very simple argument, we prove that if are nonnegative integers
then \sum_{k=0}^l(-1)^{m-k}\binom{l}{k}\binom{m-k}{n}\binom{2k}{k-2l+m}
=\sum_{k=0}^l\binom{l}{k}\binom{2k}{n}\binom{n-l}{m+n-3k-l}.
On the basis of this identity, for we construct explicit
and such that for any prime we have
\sum_{k=1}^{p-1}k^r C_{k+d}\equiv \cases F(d,r)(mod p)& if 3|p-1, \\G(d,r)\
(mod p)& if 3|p-2,
where denotes the Catalan number . For
example, when is a prime, we have
\sum_{k=1}^{p-1}k^2C_k\equiv\cases-2/3 (mod p)& if 3|p-1, \1/3 (mod p)& if
3|p-2;
and
\sum_{0<k<p-4}\frac{C_{k+4}}k \equiv\cases 503/30 (mod p)& if 3|p-1, -100/3
(mod p)& if 3|p-2.
This paper also contains some new recurrence relations for Catalan numbers.Comment: 22 page
A characterization of covering equivalence
Let A={a_s(mod n_s)}_{s=1}^k and B={b_t(mod m_t)}_{t=1}^l be two systems of
residue classes. If |{1\le s\le k: x=a_s (mod n_s)}| and |{1\le t\le l: x=b_t
(mod m_t)}| are equal for all integers x, then A and B are said to be covering
equivalent. In this paper we characterize the covering equivalence in a simple
and new way. Using the characterization we partially confirm a conjecture of R.
L. Graham and K. O'Bryant
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