Spatial coherence describes the correlation between signals at different points in space. Temporal coherence describes the correlation or predictable relationship between signals observed at different moments in time.

Spatial coherence is described as a function of distance, and is often presented as a function of correlation versus absolute distance between observation points . If , there is no spatial coherence at , i.e. .

The same operation can be performed in time with the results plotted as correlation versus relative delay . For the echo signal over the ensemble, temporal correlation is periodic at the inverse of the pulse repetition frequency (1/PRF), as shown in Figure 7.1. This holds for both transmit and receive, and for a point target or a speckle target.

The temporal correlation length of pulsed ultrasound is typically on the order of 1 s, and in the case of repeated interrogation, is periodic. Therefore the speckle target looks the same on each interrogation. For incoherent radiation, there is no temporal coherence on the order of the pulse repetition interval. All radiation has some finite correlation length.

If you capture an image within the correlation length, you get a single speckle pattern. Laser light has a temporal correlation length on the order of seconds, therefore one can observe stable speckle patterns from scattering functions illuminated with laser light. Images captured at different times with incoherent radiation are uncorrelated, and are averaged in the final image if viewed with a detector whose response time is longer than the finite correlation length.

The spatial correlation of echoes from a point target is constant. For a speckle target, the VCZ curve takes over. (If this were not the case, spatial compounding would have no effect.) Propagation is a low pass filter, therefore we expect the spatial correlation length to increase with distance from the source.

The spatial coherence of the radiation falling on a surface can be
measured by changing the spacing between the two openings in the
surface at and , and observing the interference pattern
that is generated on a screen beyond, as shown in Figure
7.2. If the pattern dies out after the first fringe, the
temporal coherence is on the order of one wavelength. For laser
light, the interference pattern is very wide. If intensity patterns
of **incoherent** radiation could be detected within its very short
temporal correlation length, then the fringe pattern would extend out
to infinity. Over many events, averaging flattens out this extended
pattern. Assuming some type of averaging is occurring, often in the
detector, only waves with some finite coherence will interfere. In
optics using an intensity detector, the number of averaged images
approaches infinity. The width of the interference pattern tells us
about the temporal and spatial coherence of the radiation.

In optics, temporal coherence is also measured by combining beams from the same source but having a known path length difference, and observing the interference pattern produced. This path length difference is achieved using a beam splitter, as shown in Figure 7.3.

The normalized correlation coefficient is the cross correlation function adjusted to remove effects related to the energy in the signals. This coefficient is often designated by the variable . Given two random variables and , the continuous time expression for is:

(7.1) |

As reviewed in Section 5.3, we can simplify this expression using that for variance. Here denotes ``expected value of'':

(7.2) |

The correlation coefficient goes as the cosine between two vectors. ``Half-way,'' the correlation has a value of 0.707 = .

The covariance definition of spatial coherence:

(7.3) |

Given stationarity:

(7.4) |

In Goodman this is referred to as the ``mutual coherence function''[12].

The covariance definition of temporal coherence:

(7.5) |

For the averaging of intensity patterns, as in compounding:

(7.6) |

As shown in Figure 7.2, given ,

(7.7) |

Consider the vector notation and for the complex instantaneous amplitudes of two signals separated in phase by the angle :

(7.8) |

Using these equations to calculate the effect of spatial compounding on the signal-to-noise ratio (SNR). The compounding of two uncorrelated images produces an SNR improvement of .

(7.9) |

In this terminology, Parseval's theorem is expressed:

(7.10) |

in which phase information has been lost through conjugate multiplication.

The power spectral density of the output of a random process is the squared modulus of the transfer function of linear system times the power spectrum density of the input random process:

(7.11) |

The cross correlation function and cross spectral density function are Fourier transform pairs:

(7.12) |

Neither contains meaningful phase information, and each contains equivalent information.

If we know the *k*-space windows of a system and target
function, or two different windows of the system under compounding,
the product of these windows gives the correlation between their echo
signals.

The cross spectrum density is a measure of the similarity of two signals at each complex frequency:

(7.13) |

In terms of two complex signals and and their transforms and ,

lim | (7.14) |

The limit as is required to generalize this expression to include functions that do not have analytical Fourier transforms.

This function is a measure of spectral similarity at each frequency:

(7.15) |

For having equal phase profiles, is a purely real number due to the conjugate operation. For unequal phase profiles, this quantity will be complex and exhibit interference patterns.

Integration of the *k*-space overlap gives the correlation coefficient
at :

(7.16) |

Envelope detection shifts the signal band to base band, losing the carrier frequency. Ideally, everything but the carrier frequency is preserved, e.g. axial and lateral bandwidth are preserved. The of the detected signals is the square of of the corresponding RF signals.