1 Introduction
^{ $\text{1}$ } Gravitational wave bands conventionally divide based on detector technology [97, 30] : Future extremelylowfrequency ( $\sim {10}^{18}$ to $\sim {10}^{15}Hz$ ) search programs will be based on mapping the intensity and polarization of the cosmic microwave background; verylow frequency observations ( $\sim {10}^{9}$ to $\sim {10}^{6}Hz$ ) mostly use pulsar timing observations; lowfrequency ( $\sim {10}^{6}$ to $\sim {10}^{1}Hz$ ) observations currently use Doppler tracking of spacecraft (perhaps a decade from now, a laser interferometer in space); highfrequency ( $\sim 1$ to $\sim {10}^{4}Hz$ ) observations involve groundbased laser interferometers or resonant bar detectors.
^{ $\text{2}$ } Although conceptually important – and used with excellent success as part of the hydrogenmaserbased suborbital GPA experiment [113] – oneway Doppler presents a practical problem for precision tracking of deep space probes: Flightqualified frequency standards for deep space are substantially less stable than groundbased standards. The quality of oneway spacecraft Doppler GW measurements is severely limited by noise in the flight frequency generator. Deep space tracking systems circumvent this by measuring twoway Doppler. In the twoway mode the ground station transmits a radio signal referenced to a highquality frequency standard. The spacecraft receives this signal and phasecoherently retransmits it to the earth. The transponding process adds noise, but at negligible levels in current observations (see Table 2 ), and does not require a good oscillator on the spacecraft.
2 Notation, Acronyms, and Conventions

3 Gravitational Wave Signal Response
$$\begin{array}{c}{y}_{2}^{gw}\left(t\right)=\frac{\mu 1}{2}\overline{\Psi}\left(t\right)\mu \overline{\Psi}\left(t\frac{1+\mu}{2}{T}_{2}\right)+\frac{1+\mu}{2}\overline{\Psi}(t{T}_{2}),\end{array}$$  (1) 
$$\begin{array}{c}{\mathbf{e}}_{+}=\left(\begin{array}{ccccc}1& & 0& & 0\\ 0& & 1& & 0\\ 0& & 0& & 0\end{array}\right),{\mathbf{e}}_{\times}=\left(\begin{array}{ccccc}0& & 1& & 0\\ 1& & 0& & 0\\ 0& & 0& & 0\end{array}\right).\end{array}$$  (2) 
$$\begin{array}{c}{y}_{2}^{gw,LWL}\left(t\right)\to \frac{{T}_{2}}{2}({\mu}^{2}1){\overline{\Psi}}^{{}^{\prime}}\left(t\right).\end{array}$$  (3) 
4 Apparatus and Principal Noise Sources
^{ $\text{3}$ } Noises are characterized in the time domain by Allan deviation, ${\sigma}_{y}\left(\tau \right)$ , or in the frequency domain by the power spectra of fractional frequency fluctuations, ${S}_{y}\left(f\right)$ . These are related by
4.1 Frequency standard noise
4.2 Plasma scintillation noise
^{ $\text{4}$ } There is a large literature on wave propagation through random media. Excellent general references for radiowave propagation observations include [95, 71, 33, 88, 61] .
4.3 Tropospheric scintillation noise
4.4 Antenna mechanical noise
^{ $\text{5}$ } The mechanical stability of the DSN's 70m antennas has not been systematically studied. A very few observations done with Cassini in 2003 suggest mechanical noise of the 70m antennas is substantially larger than for the 34m beamwaveguide antennas.
4.5 Ground electronics noise
4.6 Spacecraft transponder noise
^{ $\text{6}$ } In principle, Cassini has one transponder and one translator. The distinction is that a transponder performs functions in addition to phasecoherent generation of the downlink signal from the uplink signal.
4.7 Thermal noise in the ground and spacecraft receivers
4.8 Spacecraft unmodeled motion
4.9 Aggregate spectrum
4.10 Summary of noise levels and transfer functions
$$\begin{array}{ccc}{y}_{2}^{noise}\left(t\right)& =& {y}^{FTS}\left(t\right)*\left[\delta \right(t)\delta (t{T}_{2}\left)\right]+{y}^{sw}\left(t\right)*\left[\delta \right(t)+\delta (t{T}_{2}+2x/c\left)\right]+\end{array}$$ 
$$\begin{array}{ccc}& & {y}^{ion}*\left[\delta \right(t)+\delta (t{T}_{2}\left)\right]+{y}^{s/celect}\left(t\right)*\delta (t{T}_{2}/2)+{y}^{s/cmotion}*\delta (t{T}_{2}/2)+\end{array}$$ 
$$\begin{array}{ccc}& & {y}^{ant}\left(t\right)*\left[\delta \right(t)+\delta (t{T}_{2}\left)\right]+{y}^{tropo}\left(t\right)*\left[\delta \right(t)+\delta (t{T}_{2}\left)\right]+{y}^{groundelect}\left(t\right)+\end{array}$$ 
$$\begin{array}{ccc}& & {y}^{rcvr}+{y}^{systematic}\left(t\right),\end{array}$$  (4) 
$$\begin{array}{ccc}{y}_{2}\left(t\right)& \simeq & {y}_{2}^{gw}+{y}^{FTS}\left(t\right)*\left[\delta \right(t)\delta (t{T}_{2}\left)\right]+{y}^{s/celect}\left(t\right)*\delta (t{T}_{2}/2)+\end{array}$$ 
$$\begin{array}{ccc}& & {y}^{s/cmotion}*\delta (t{T}_{2}/2)+{y}^{ant}\left(t\right)*\left[\delta \right(t)+\delta (t{T}_{2}\left)\right]+{y}^{tropo}\left(t\right)/10*\left[\delta \right(t)+\delta (t{T}_{2}\left)\right]+\end{array}$$ 
$$\begin{array}{ccc}& & {y}^{groundelect}\left(t\right)+{y}^{rcvr}+{y}^{systematic}\left(t\right)\end{array}$$ 
$$\begin{array}{ccc}& =& {y}_{2}^{gw}+{y}_{2}^{other}\left(t\right),\end{array}$$  (5) 

5 Signal Processing
5.1 Noise spectrum estimation
^{ $\text{7}$ } The bispectrum is the Fourier transform of the thirdorder lagged product $\langle x\left(t\right)x(t+{\tau}_{1})x(t+{\tau}_{2})\rangle $ ; it gives the contribution to the third moment from the product of three Fourier components having frequencies which add to zero. It has been used in many fields, notably geophysics, to study weak nonlinearities (see, e.g., [49, 72] ).
5.2 Sinusoidal and quasiperiodic waves
5.3 Bursts
5.4 Stochastic background
5.5 Classification of data intervals based on transfer functions
5.6 Frequencytime representations
5.7 Qualifying/disqualifying candidates
5.8 Other comments
^{ $\text{8}$ } An exception was with Mars Observer [56, 12] where spacecraft engineering telemetry was crucial for correcting the Doppler for the (slow) spacecraft spin.
6 Detector Performance
6.1 Observations to date

6.2 Sensitivity to periodic and quasiperiodic waves
6.2.1 Sinusoidal waves and chirps
^{ $\text{9}$ } This was suggested by Estabrook as giving “noncommittal” directions on the celestial sphere and have separations which are reasonably matched to the effective angular response of a typical Doppler tracking observation.
6.2.2 Nonsinusoidal periodic waves
6.3 Burst waves
6.4 Sensitivity to a stochastic background
7 Improving Doppler Tracking Sensitivity

^{ $\text{10}$ } Signal processing procedures which exploit differences in the signal and noise transfer functions can give improved sensitivity at selected frequencies (see, e.g.,[99, 8, 10] ).
8 Future LowFrequency Detectors
^{ $\text{11}$ } Oneway tracking emphasizes the symmetry of the LISA array and simplifies the analysis of the apparatus; for technical reasons the actual implementation of LISA may involve some of the links being twoway [91] .
^{ $\text{12}$ } TDI developed in increasing sophistication to account for unequal armlengths, differences between the (unequal) armlengths on given upand down links due to aberration, and timedependences of the unequal, aberrated armlengths. For a discussion of this development see [104] and references therein. TDI also allows LISA's laser noises to be canceled in many ways [14, 105] . In particular, one lasernoisefree combination is insensitive to GWs, but responds to the instrumental noises; this combination will be used to discriminate a stochastic GW background due to galactic binary stars from instrumental noises [103, 38] . Because multiple lasernoisefree combinations can be simultaneously constructed, the optimum sensitivity of the LISA array can be achieved by appropriately linear combinations of the TDI data streams [82] .
9 Concluding Comments
10 Acknowledgements
Note: The reference version of this article is published by Living Reviews in Relativity