**Quadrupole BPMs at the Accumulator.**

Beam position detectors are the most popular tool in beam diagnostics.
Most frequently used ones are electrostatic pick-ups sensing beam
position in one of the transverse planes. Quadrupole BPM combines both x
and y sensors at the same location. As it will be shown further, not
only does this allow to do measurements in each plane, but also gains in
ability to measure important nonlinear effects such as quadrupole
oscillations.

Accumulator quad BPM is made of 2 pairs of electrostatic plates, see
Figure 1. Plates are 1 m long and 1 cm wide. Distance between opposite
plates is 2.27 cm.

![](media/image1.jpeg){width="1.3520833333333333in"
height="1.2534722222222223in"}

**Figure 1**. Geometry of the Accumulator Quadrupole pickup.

Plates are connected to the preamplifiers sitting right there on the
vacuum chamber, see Figure 2. There are two such devices installed next
to each other in sector 60, the normal one and the scew QBPM, the latter
has sensing plates rotated 45 degrees around the beam axis.

The whole devices are mounted on the moving XY tables and their position
may be controlled remotely with precision of better than 0.1 mm. The
functions at the location of the normal QBPM are bx=24m, by=4m, Dx=0.26.
Signals from each of 4 PA are sent via twisted pair lines up to building
AP10 and connected there to the DATEL ADC card (4channels, PCI, 14 bit,
10MHz). The scew QBPM has not been used in these studies yet.

![](media/image2.jpeg){width="3.667361111111111in"
height="2.4444444444444446in"}

**Figure 2**. The QBPM in straight section 60 of Accumulator ring.

**Signal combinations**

A simple single pair pickup response is never exactly linear with the
beam displacement. Leaving only first and second order terms we have:

$V_{x \pm}(x,y) = \text{ZI}(t)\left\{ 1 \pm \text{ax} + b(x^{2} - y^{2}) \right\}$
(1a)

$V_{y \pm}(x,y) = \text{ZI}(t)\left\{ 1 \pm \text{ay} - b(x^{2} - y^{2}) \right\}$
(1b)

where *a* and *b* are the pickup geometry parameters. Using 4 signals
one can build combinations:

> $\text{DX} = X2 - X1 = V_{x +} - V_{x -} \propto a\left\langle x \right\rangle$
> (2a)
>
> $\text{DY} = Y2 - Y1 = V_{y +} - V_{y -} \propto a\left\langle y \right\rangle$
> (2b)
>
> $$\text{SX} = X2 + X1 \propto 1 + b\left\langle x^{2} - y^{2} \right\rangle$$
>
> $\text{SY} = Y2 + Y1 \propto 1 - b\left\langle x^{2} - y^{2} \right\rangle$
> (3)

Combinations (2) are normally used in regular dipole pickups.
Combinations 3 are not of a good use unless both are available at the
same location. In case of a quadrupole pickup they are and combination

> $Q = \text{SX} - \text{SY} \propto b\left\langle x^{2} - y^{2} \right\rangle$
> (4)

gives access to the nonlinear term of pickup response.

Ellipticity $\left\langle \sigma_{x}^{2} - \sigma_{y}^{2} \right\rangle$
can be used to determine the beam size or to observe quadrupole
oscillations of the beam size. More details about the technique will be
given in following section.

**Betatron mismatch.**

The primary purpose of QBPM is measuring the betatron mismatch between
the AP3 beam line and the Accumulator. In normalized coordinates

$Y = \frac{x}{\sqrt{\beta}}$

$Y' = \frac{\alpha x}{\sqrt{\beta}} + x'\sqrt{\beta}$

The phase diagram of the beam is represented by the circle as shown in
Figure 3 (the dash-dotted line). Beam emittance is the area under this
circle and is equal to (π times) square radius A. If beam coming from
the beamline has emittance A^2^ and both beta functions and its
derivatives at the end of line and at injection point in the ring are
equal, the phase diagram of this beam is the same circle and continues
to be so as beam continues coasting in the ring.

If functions are not matched between beamline and the ring, we find its
phase diagram being an ellipse (red trace). As moving in the ring all
particles will rotate in circles around the center of the diagram with
betatron frequency, and so will the whole ellips. It is clear in this
case that the size of the beam will experience oscillations at double
the betatron frequency (quadrupole oscillations). This holds only as
long as motion of individual particles can be considered synchronous.
That is not true for infinitely long time because each particle has its
own betatron frequency, and eventually ellipse smears into a circle with
radius equal to the largest half size of this ellipse. The final
emittance becomes

**Figure 3.** Matched and mismatched phase diagrams of injected beam.

$\varepsilon = \varepsilon_{0} \ast H$, (5)

where
$H = \frac{1}{2}\left\{ \frac{\beta_{1}}{\beta_{2}} + \frac{\beta_{2}}{\beta_{1}} \right\} + \frac{1}{2}\left\{ \left\lbrack \alpha_{1} - \alpha_{2}\frac{\beta_{1}}{\beta_{2}} \right\rbrack^{2}\frac{\beta_{2}}{\beta_{1}} \right\}$,

$\beta_{1}$, $\alpha_{1}$- functions in the end of the beamline

$\beta_{2}$, $\alpha_{2}$- functions at the injection point in the ring

parameter $\lambda$ determines how much ellipse is close to the circle,
see Figure 3.

$$\lambda = \frac{\sqrt{H + 1} + \sqrt{H - 1}}{\sqrt{2}}$$

The amplitude of beam size oscillations is
$\sigma_{{0x}^{2}}\sqrt{\frac{\Delta\varepsilon}{2\varepsilon_{0}}}$

Initial ellipse is described by

$Y^{2}\overset{\sim}{\gamma} + Y'^{2}\overset{\sim}{\beta} - 2\text{YY}'\overset{\sim}{\alpha} = A^{2}$
(6)

where
$\overset{\sim}{\alpha} = \frac{\beta_{1}}{\beta_{2}} + \left\lbrack \alpha_{1} - \alpha_{2}\frac{\beta_{1}}{\beta_{2}} \right\rbrack^{2}\frac{\beta_{2}}{\beta_{1}}$,
$\overset{\sim}{\beta} = \frac{\beta_{2}}{\beta_{1}}$,

Initial angle of ellipse can be determined from the phase of quadrupole
oscillations. Combined with the amplitude of oscillations, it gives an
unambiguous determination of α and β mismatch.

Two parameters are measured for quadrupole oscillations: the phase and
the amplitude, this determines $\Delta\alpha$ and $\Delta\beta$.

**Frequency content of the signal**

Consider transverse motion (let's take horizontal plane) of an
individual particle. Particle position is a sum of 3 components:

> $$x_{i} = X_{0} + X_{\beta} + {\overset{\sim}{x}}_{i}$$

X~0~ - orbit position ,

$X_{\beta}(t) = X_{d}\text{cos}(\omega_{r}\nu_{x}t + \phi_{x})$ -
coherent dipole motion ,

*x~i\ ~*-individual betatron motion with respect to the center of beam.

Putting this into (1a) and taking average over all particles in the beam
we have

$V_{x \pm}(t) = Z\text{*I}(t)$ (7)

$1 + b \ast (X_{0^{2}} - Y_{0^{2}}) + 0\text{.}5b \ast (X_{d^{2}} - Y_{d^{2}}) \pm \text{aX}_{0} + b \ast (\sigma_{{0,x}^{2}} - \sigma_{{0,y}^{2}}) +$
(I)

$\pm \text{aX}_{\beta}(t) + 2b \ast (X_{0}X_{\beta}(t) - Y_{0}Y_{\beta}(t)) +$
(II)

$b \ast (X_{\beta^{2}}(t) - Y_{\beta^{2}}(t)) + b \ast (\sigma_{{2,x}^{2}}(t) - \sigma_{{2,y}^{2}}(t))$
(III)

This expression is broken up into 3 parts for convenience. "$\pm$" here
corresponds to left and right (top and bottom) plates. Picking right
signal combination one can effectively cancel contribution of symmetric
or asymmetric components. Part (7-I) contains constant parameters,
therefore it appears in signal as constant term and revolution
harmonics. Looking at *S0=SX+SY* at revolution frequency and moving the
beam center position one can find the value of *b* and the beam center
position. Plotting *DX* the same way would give value of *a*.
Combination *Q* can even be used to measure the beam size if electronics
is calibrated at the level of better than 0.1%.

Part (7-II) contains linear amplitudes and mainly appears at betatron
frequencies. *DX* and *DY* should give a clear dipole signal at betatron
frequency.

Consequently, part (7-III) works at double betatron frequencies. It is
clear that quadrupole signal combination *Q* is contributed by two
sources, and in order to get a clear quadrupole oscillation one needs to
either make the contribution of dipole motion negligibly small, or be
able to subtract it from the full quadrupole signal.

**Is subtraction necessary?**

Presently the requirement posed on beam steering at injection is
determined by mainly the operational constraints and is defined as
"keeping the steering parameter below 0.4". Translated into the dipole
oscillations amplitude at the location of the QBPM it is
$X_{d} \leq 0\text{.}4 \ast \sqrt{\frac{\beta_{x}}{3}} = 1\text{.}1\text{mm}$.

This results in emittance dilution of
${\Delta\varepsilon}_{n} = 3\gamma_{c}(\beta_{x}{\Delta\theta}^{2} + 2\alpha_{x}\Delta\theta\Delta x + \gamma_{x}{\Delta x}^{2})$

$\Delta x = X_{d}$, ${\Delta\varepsilon}_{n} \leq$`<!-- -->`{=html}2.0π,
which is quite substantial amount and may be addressed in future. Until
then, however, pure quadrupole oscillation signal may be observed
without subtraction of the dipole component if it is bigger than that,
or in our case,

bigger than $0\text{.}6\text{mm}^{2}$. This would correspond to betatron
emittance dilution of

$\frac{\Delta\varepsilon}{\varepsilon} =$`<!-- -->`{=html}0.04 for
$\varepsilon_{0} = \text{10}\pi$ ,

hence limiting betatron mismatch measurement at
$\frac{\Delta\varepsilon}{\varepsilon} =$`<!-- -->`{=html}10-15%.

**More about a signal of real beam**

If V(t) is the signal induced on the pickup plates, then the end signal
received from the DAQ is its convolution with the response function of
the electronics

$$U(t) = V(t) \otimes G$$

In frequency domain, we will present G(w) for convenience, as G=G~0~g(w)
with g(w) being the frequency response function and G~0~ - the general
scale constant. For further convenience we will hide some other
constants into G~0~, too. Remember that V(t) is a function of *t*
described in eq.(7) Let's consider part (7-I) in the simplest form
$V(t) = Z\text{*I}(t)$, and I(t) is a periodic function
$I(t) = I_{0} \ast \sum_{}^{}{f(t - \text{Tn})}$, where $f(t)$ is a
rectangular function. Also taking in mind that every particle has its
own revolution frequency, one can easily find out that the end signal
will look like

$U(t) = g(\omega_{0}n)G_{0}\overline{I}\left\{ 1 + 2\sum_{n = 1}^{\infty}{\left\lbrack \frac{\text{sin}(\frac{\omega_{0}n\tau}{2})}{(\frac{\omega_{0}n\tau}{2})} \right\rbrack \ast \left| F(\frac{n\eta}{P_{0}}\omega_{0}t) \right|\text{cos}(\omega_{0}\text{nt} + \phi_{0})} \right\}$
(8)

*F(x)* here is the Fourier transform for the momentum distribution
function. For terms like (7-II):

$V(t) = g(\omega_{r}\nu_{x})G_{0}I(t)\text{*c}os\left\lbrack \omega_{r}\nu_{x}t + \phi_{x} \right\rbrack$


$U_{n}(t) = g(\omega_{0}\left| \nu_{0} \mp n \right|) \ast G_{0}\overline{I} \ast \left\lbrack \frac{\text{sin}(\frac{\omega_{0}n\tau}{2})}{(\frac{\omega_{0}n\tau}{2})} \right\rbrack\text{cos}\left\lbrack \omega_{0}(\nu_{0} \mp n)t + \phi_{x} + \delta_{n, \pm} \right\rbrack \ast \left| F\left( \frac{C_{x} - \eta(\nu_{0} \pm n)}{P_{0}}\omega_{0}t \right) \right|$

\(9\)

where

$F\left( \frac{C_{x} - \eta(\nu_{0} \pm n)}{P_{0}}\omega_{0}t \right) = \left| F\left( \frac{C_{x} - \eta(\nu_{0} \pm n)}{P_{0}}\omega_{0}t \right) \right| \ast e^{{i\delta}_{n, \pm}}$
,

Here we had to take into account chromaticity Cx. And similarly,
decomposition into series for functions of the (7-III) type is

$V(t) = g(\omega_{r}\nu_{x}) \ast G_{0}I(t)\text{*cos}^{2}(\omega_{r}\nu_{x}t + \phi_{x})$


$$U_{n}(t) = \frac{1}{2}G_{0}\overline{I} \ast \left\lbrack \frac{\text{sin}\left( \frac{\omega_{0}\text{nτ}}{2} \right)}{\left( \frac{\omega_{0}\text{nτ}}{2} \right)} \right\rbrack$$

$+ g(\omega_{0}\left| 2\nu_{0} \mp n \right|)\left| F\left( \frac{2C_{x} - \eta(2\nu_{0} \pm n)}{P_{0}}\omega_{0}t \right) \right|\text{cos}\left( \omega_{0}(2\nu_{0} \mp n)t + 2\phi_{x} + \delta_{2,n, \pm} \right)$
(10)

> Obvious properties of *F(x)*:

$\begin{array}{r}
F(x) \simeq 1, \\
\delta_{2,n, \pm} \simeq 0
\end{array}$ when *x* \<\< $\frac{1}{\Delta p}$

$\delta_{2,n, \pm} = 0$ if *f(p)* is symmetric.

It can be clearly seen that at high chromaticity signal decays quickly
due to beam decoherence. Another conclusion from equations 8-10 is that
signal is strongly dominated by the common mode component (8). In order
to suppress revolution harmonics in the signal, front end preamplifiers
had been designed as the bandpass filters with maximum response at 90
kHz and 30 dB suppression at f~0~.

**Beam position based calibration**

If beam is well centered in the pick-up, X~0~=0, Y~0~=0, then the common
mode signal induced at all 4 plates is equal. This fact can be used to
match the gains in channels *g~x1~*, *g~x2~*, *g~y1~*, *g~y2~*. One must
however first determine this beam center position. This can be done with
the position scans thanks to the moveable stands holding the QBPMs.

Figure 4 shows the horizontal position scan. Ideally parabola minimum
would indicate position X~0~=0, and DX line would cross 0 at this point
if gains *g~x1~*, *g~x2\ ~* are matched. However parabola minimum is not
located exactly at X~0~=0 if gains are not matched. Therefore we used
iterative process: from DX at the position of parabola minimum both
mismatch and parabola offset were estimated, correction was made to the
electronics response and then curves were recalculated again. This
process quickly converges and yields values of gain correction and
geometric parameters *a* and *b*. Matching the gain between horizontal
and vertical signals can be done requiring parabola minimum be equal to
1.

![](media/image3.jpeg){width="4.125in" height="3.877083333333333in"}

**Figure 4.** Beam position scan. *a*~0~=0.0498,    *b*~0~=0.00132

The limitation of this method is that it gives both gain matching and
geometric constants at the revolution frequency only. As a side output
of these scans distance between core and extraction orbits was measured
on shot lattice:  

$X_{0,\text{core}} - X_{0,\text{extr}} =$`<!-- -->`{=html}3.5 mm , which
leads to $D_{x} =$`<!-- -->`{=html}0.26 m at the location of QBPM. In
order to match gains at arbitrary frequency we were applying the
generator signal to one of the channels while measuring response on
orthogonal plates. Although this turned out to be an efficient and
straightforward way of cross-calibration, it still lacks the absolute
gain calibration and it suffers at revolution frequency because of
filtering in preamplifiers.

**Calculation of geometric parameters and their frequency dependence.**

In case of infinite preamplifier input impedance beam with charge Q~b~
would induce potential at one of the plates

$\phi_{i\infty}(x,y) = a(i,x,y) \ast Q_{b}$

and when beam is exactly at the center, $\phi_{i\infty}(0,0) = V_{0}$

for small compared to aperture deviations of beam from the center,
induced potential may be approximated as

> $$\phi_{i\infty}(x,y) = V_{0} \ast \left\{ 1 \pm a_{0} \ast x + b_{0} \ast (x^{2} - y^{2}) \right\}$$

geometric coefficients$a_{0}$, $b_{0}$ can be computed from the
electrostatic model of BPM.

This was done using MERMAID program. Obtained response signal
combinations DX and Q are shown in Figure 5 as functions of pick-up
position.

![](media/image4.wmf)

**Figure 5**. Induced signal on the plates as function of PU position.

Now let's take into account the preamp's impedance at finite frequency,
see Figure 6.

![](media/image5.jpeg){width="1.538888888888889in"
height="0.8638888888888889in"}

**Figure 6.** Input impedance of the QBPM preamplifier.

as $C_{A} \simeq C_{d}$, effective parallel capacitance significantly
changes at low frequencies.

> $\phi_{i,w}(x,y) = \phi_{{i,}_{\infty}}(x,y) + S_{\text{ik}} \ast q_{k}$,

where
$\left\lbrack S_{\text{ik}} \right\rbrack = \left\lbrack C_{\text{ik}} \right\rbrack^{- 1}$,
inverse capacitance matrix. Solving this equation yields:

$\phi_{i,w}(x,y) = \left\lbrack \delta_{\text{ik}} + \frac{C_{A}}{1 + i\omega\text{RC}_{A}}S_{\text{ik}} \right\rbrack^{- 1}\phi_{k_{\infty}}(x,y)$
(11)

and coefficients ***a***   and  ***b***  become frequency dependent.

The capacitance matrix was calculated in framework of the same program.
Setting on the right plate such a charge that its potential is 1V (this
is not necessary, and was done for convenience of computations only),
see Figure 7, we calculate potential on other plates.

![](media/image6.jpeg){width="1.6083333333333334in"
height="1.6083333333333334in"}

**Figure 7.** Computation of the cross-capacitance matrix.

$$S_{i} = \frac{V_{i}}{q_{0}} = \begin{bmatrix}
5\text{.}\text{708} \cdot \text{10}^{- 3} \\
0\text{.}\text{04625} \\
0\text{.}\text{1582} \cdot \text{10}^{- 3} \\
0\text{.}\text{1582} \cdot \text{10}^{- 3}
\end{bmatrix}$$

Now we can reconstruct the whole matrix using its symmetry:

$$S_{\text{12}} = S_{\text{21}} = S_{\text{34}} = S_{\text{43}} = \text{Si}_{1}$$

$$S_{\text{13}} = S_{\text{31}} = S_{\text{14}} = S_{\text{41}} = S_{\text{32}} = S_{\text{23}} = S_{\text{42}} = S_{\text{24}} = \text{Si}_{3}$$

$$S_{\text{11}} = S_{\text{22}} = S_{\text{33}} = S_{\text{44}} = \text{Si}_{2}$$

and now using eq. (11) we can calculate dependence ***a***(f) and
***b***(f), as shown in Figure 8.

![](media/image7.wmf)

![](media/image8.wmf)

**Figure 8**. Calculated dependence ***a***(f) and ***b***(f)

The measured and calculated parameters are compared in Table 1. There is
a very good agreement with the horizontal scan measurement. For reasons
not yet fully understood at this moment vertical scan doesn't agree very
well with the horizontal one.

**Table 1.** The calculated and measured geometric parameters.

  --------- ---------------- -------------- --------------- ----------------
            Model, f=        Model, f=f0    Horz. meas.     Vert. meas.
            $\infty$                                        

  ***a***   0.0450           0.0493         0.0498          0.0455

  ***b***   0.00119          0.00135        0.00132         0.00146
  --------- ---------------- -------------- --------------- ----------------

$$$$

**Measurement of the quadrupole oscillations at lowered chromaticity.**

As it has been pointed out earlier, quadrupole oscillations decohere
quickly if the chromaticity is high. Unfortunately, this is the case for
the extraction orbit in Accumulator. It is more important to keep
chromaticity low on the core orbit where the beam lives for many hours
rather than at extraction, where beam normally spends only minutes
before being transferred to MI. With present range of multipole magnet
strengths it is not possible to adjust chromaticity at both orbits at
the same time.

Decoherence time for dipole and quadrupole motions is of the order of
tens of revolutions. At present observation of such quick processes at
injection is obscured by to phenomena: 1) charging the pick-up plates
due to beam losses and slow dissipation of this charge, and 2)
transition signal appearing as a result of step-like change in DC beam
current. The lower filter band frequency, the longer this transition
lasts. Both these issues are being addressed at the moment, in the
meantime we made a dedicated measurement with lowered down chromaticity.
Chromaticity at extraction orbit can be reduced by lowering current in
octupole A:OCT10 by 40 Amps. This study was done without stack because
during this time chromaticity at core orbit was high.

For the beginning we set up injection with a large dipole mismatch, so
that the quadrupole signal is completely dominated by the dipole
component, in order to see if we can successfully determine this
component based on the dipole signal measured at single betatron
frequency.

We can rewrite amplitudes of known signal combinations according to eq.
(8-10):

$S_{0} = 8g_{0}G_{0}\overline{I}\frac{\text{sin}\left( \frac{\omega_{0}\tau}{2} \right)}{\left( \frac{\omega_{0}\tau}{2} \right)}$
(12)

$\text{DX}_{0^{(1)}} = \frac{a_{1}}{4}X_{d}\frac{g_{1}}{g_{0}}S_{0}$
(13)

$Q_{0^{(2)}} = \frac{b_{2}}{4}\frac{g_{2}}{g_{0}}S_{0}X_{d^{2}} = 4\frac{b_{2}}{a_{1^{2}}}\frac{g_{2}g_{0}}{g_{1^{2}}}\frac{\text{DX}_{0^{2}}}{S_{0}}$
(14)

where $a_{1} = a(\omega_{0}\left( 1 - \nu_{x} \right))$,
$b_{2} = b(\omega_{0}\left( 2\nu_{x} - 1 \right))$, upperscript (1) and
(2) in eq.(13) denote single and double betatron frequency. It seems
convenient to tie amplitudes with parameter *K~f~*, which can be
independently determined from both measurements and calculations:

$\frac{\text{DX}_{0^{(1)}}}{S_{0}} = X_{d} \cdot K_{f}$,
$K_{f} = \frac{a_{1}}{4} \cdot \frac{g_{1}}{g_{0}}$

$$Q_{0^{(2)}} = \frac{b_{2}}{a_{1}}\frac{g_{2}}{g_{1}}\frac{1}{K_{f}}\frac{\text{DX}_{0^{2}}}{S_{0}}$$

using parameters ***a**, **b*** calculated in previous sections, we
expect the dipole term in the quadrupole signal to have the form:

$$\text{Qfc}(t) = Q_{0}^{(2)}e^{- \frac{2t^{2}}{\sigma^{2}}}\text{cos}\left( 2\pi \cdot f_{0}(2\nu_{x} - 1)t + \varphi_{1} + \varphi_{\text{PA}} \right)$$

Here time decay constant $\sigma$ and phase $\varphi_{1}$ are taken from
the dipole mode fit, and $\varphi_{\text{PA}}$ is the phase advance
between single and double betatron frequency in the preamp response.

![](media/image9.wmf)

**Figure 9**. Measured and calculated dipole component in quadrupole
signal

![](media/image10.wmf)

**Figure 10**. Pure quadrupole oscillation signal in absence of dipole
component.

Both amplitude and phase of measured quadrupole signal agree very well
with expected waveform calculated from the dipole signal.

Different beamline tuning when in contrary, steering error is very
small, is shown in Figure 10. The dipole component is negligible and
quadrupole oscillation signal is clearly seen although its amplitude is
low. Parasitic components at revolution and betatron frequencies were
removed from the *Quad_t* signal, this was done by filtering a broad
region around double betatron frequency in FFT spectrum of Q signal
combination and then deconverting this region back to time domain. This
explains aliasing modulations of the signal sin-wave. Observed amplitude
of quadrupole oscillation is consistant with the introduced betatron
mismatch assuming $\varepsilon_{\text{beam}} = \text{16}\pi$ for
injected Reverse proton beam, and corresponds to emittance growth of
50%. ADC sensitivity is not a limiting factor, however, because gain may
be increased by a factor of 50 in preamplifier without yet reaching the
limit of dynamic range in ADC.

Thermal noise of an input circuit is equivalent to 1/50 ADC LSB in 200
kHz band, and is not an issue without substantial increase of bandwidth.

**Conclusions**

In the studies performed with Quadrupole BPM at Accumulator ring it was
demonstrated that quadrupole oscillation modes may be observed and
measured using this instrument if chromaticity is low at extraction
orbit. Mathematical model of the pick-up response and signal development
shows reasonable agreement with the collected data.

In order to make this device serve operations, however, electronics
needs to be modified in order to recognize quick developing signals.