# Frequency encoding We have now selected a slice by applying a gradient in the z-axis. Should we want to select a section of the slice in the x-axis (i.e a column), all nuclei along the x-axis of the slice will have different amplitudes (indicating different brightness values) but they will have the same frequency and phase. Adding all the signals together results in one large wave of the same frequency. This is of no use if we want to localise the signal in the x-axis as all locations will have the same frequency. We overcome this by applying another magnetic gradient in the x-axis. This changes the Larmor frequencies of the nuclei in a gradient along the x-axis. Each segment will now return a signal of a different frequency depending on its location along the slice. As they are of different frequencies, they will eventually become of different phases. Adding the signals together gives a large signal at the start, when they are still all in phase, but this signal drops off as the phases diverge. One of the consequences of the gradient is that as the signals go out of phase the total signal becomes very small leaving a small amount of time in which a useful signal can be measured. To overcome this, a “dephase” gradient is first applied.

The read-out gradient then rephases the MR signals such that they all come back into phase and form the maximum signal during the data collection period – the gradient echo.

Usually, the dephasing and read-out gradients are designed so that the gradient echo occurs in the middle of the data collection period.

### Decoding the signal

Now we have one MR signal formed of many MR signals of different frequencies. Luckily, we can easily extract out each frequency and its amplitude mathematically using the Fourier Transformation (FT).

We can now map each signal to its location in the x-axis of the slice by its frequency and assign the corresponding amplitude (brightness).

## Artefacts from frequency encoding

#### Aliasing The wave signal we receive has to be digitised before the FT can be applied and the frequencies extracted via sampling. If the signal is not sampled regularly enough, we will underestimate its frequency – this is called aliasing. The sampling frequency required to give an accurate result can be calculated with the Nyquist limit i.e. the maximum signal frequency that can be accurately sampled:

Nyquist limit = sampling frequency / 2

A high frequency signal is wrongly sampled as a low frequency signal and slotted into the low frequency location resulting in wrap-around.

To ensure that this limit is not surpassed the range of frequencies is limited prior to sampling using a ‘band-pass filter‘ that will only allow through a certain range of frequencies – the receiver bandwidth. The edges of the field-of-view in the frequency-encoding direction correspond to limits of the receiver bandwidth.

##### Overcoming aliasing
• Larger field of view (FOV)
• Use pre-saturation bands on the areas outside the FOV to null the signal
• Anti-aliasing software
• Switch phase and frequency directions
• Use surface coil – these are sensitive only to the areas in the FOV and also improve signal-to-noise ratio

#### Chemical shift In frequency-encoding we assume that all nuclei will have the same Larmor frequency. However, nuclei in fat and water will have slightly different frequencies due to the local magnetic field effects. This difference in frequencies due to different environments is misregistered as differences due to location.

This artefact only occurs in the frequency-encoding direction.

##### Factors affecting chemical shift
• Chemical shift increases with magnetic field strength
• Relatively greater difference between the frequencies
• Chemical shift increases with decreasing gradient strength
• Shallower gradient means more frequencies coded within the same area, small differences in frequencies will be more evident
• Narrower bandwidth gives higher chemical shift
• Same reasoning as above

### Σ  Summary

1. Gradient applied in z-axis to select axial slice
2. Dephase gradient applied along x-axis