Real v Imaginary Signals

What is the difference between real and imaginary signals? How can one be more real than another?  

Quadrature detection of the MR signal

As described in a previous Q&A, the MR signal is an induced current generated by precession of the net magnetization (M) after stimulation by an RF- pulse. The signal is commonly detected in quadrature using receiver coils sensitive to magnetic flux in two orthogonal directions. Output channels, denoted I and Q (for "in phase" and "quadrature" respectively), send their respective signals along separate digitization and amplifier pathways. These signals are ultimately demodulated, processed, and recombined to create the final MR image. 

The quadrature receiver coils are measuring the same precessing magnetization (M) from two different perspectives. The signals in the I and Q channels, therefore, should theoretically be identical except for a 90º-phase shift between them. The second coil permits knowledge of the exact position of M and hence its direction of its rotation (i.e., positive vs negative frequency). Another way to think of this is that with quadrature detection you are "listening" to the MR signal in "stereo". The same "music" is coming out of each speaker but you can now tell that the violins are on the left and the horns are on the right.

In reality the signals from the I and Q channels are not phase-shifted exact copies of one another because they also contain noise. Unlike the signals, noise in the two channels is independent and uncorrelated. Thus quadrature detection offers an increase in signal-to-noise by a factor of √2 = 1.41 over detection by a single linear receiver coils. 

The MR signal can be represented as a vector with real (Re) and imaginary (Im) components recorded from the I and Q channels respectively. An equivalent/alternative representation of the signal is as a complex number

Signal = (Re, Im) = Re + i Im

where i²  = −1, the imaginary unit. Its magnitude and phase can be calculated by simple trigonometry.

Representation of the MR signal in terms of real and imaginary components.

The designation of the I and Q signal channels as "real" and "imaginary" is entirely arbitrary.  The signal from one channel is no more or less "real" than that from the other channel.  

The signal data may be reconstructed in several ways: (1) as a "real" image, (2) as an "imaginary" image, (3) as a magnitude image, or (4) as a phase image. For illustrative purposes I have processed the data from the I and Q channels separately to demonstrate the contribution of each component more concretely.  The horizontal bands are phase-shifted from each other by 90° and reflect a baseline phase error across the imaged volume.  

Real

Imaginary

Magnitude

Phase

In clinical practice, separate I and Q channel images are never obtained, with magnitude images being used nearly exclusively for diagnosis. Phase-images are occasionally generated in clinical MRI for the depiction of flow and characterization of susceptibility-induced distortions.

Advanced Discussion (show/hide)»

Additional notes concerning non-quadrature receiver coils

Some single simple loop coils used in MR imaging are linearly polarized, not quadrature. This means that they only record a single channel of MR signal initially. I and Q channel signals can still be created from this single source, however, by splitting the signal into two parts and phase-shifting one by 90° using digital or analog techniques. This allows real and imaginary data to be input into the array processor, the format expected for standard Fast Fourier Transformation (FFT). No √2 improvement in signal-to-noise is achieved, however, since the same signals and noise are present in each channel.

Additional notes on signal-to-noise

It is generally assumed that the noise distribution in each channel is stationary, uncorrelated, and "white", having a Gaussian probability distribution. After inverse Fourier transformation of the complex data, the noise is still Gaussian.

A common property of Gaussian distributions (proved in introductory college probability theory courses) is that the sum of two Gaussian variables each with standard deviations σ is also Gaussian but with standard deviation σ(√2). Hence when the signals and noise from the I and Q channels are combined, the total signal-to-noise ratio is increased by a factor of (√2) = 1.41.

This simple analysis is not quite right when magnitude images are considered, however, as the noise probability distribution is no longer strictly "white". Since the magnitude is a sum of squares, the noise now has a Rician distribution. As the signal-to-noise approaches zero, the Rician distribution converges to a Rayleigh distribution. If the SNR is higher than about 2 then the noise probability distribution is approximately Gaussian again. In most MR imaging applications the SNR is in this higher range, allowing the Gaussian approximation to hold.

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References

     Gudbjartsson H, Patz. The Rician distribution of noisy MRI data. Magn Reson Med 1995; 34:910-914.
     Stormont RS, Anas MC, Pelc NJ. Radio frequency receiver for a NMR instrument. US Patent #4992736A, published 12 Feb 1991.     

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Related Questions
     What is the difference between linearly polarized (LP) and a circularly polarized (CP) coils?  

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