# Complex random vector
A **complex random vector** can be described either as a random element taking values in a complex vector space of finite dimension, or, equivalently, as a set of random variables with complex values whose distribution is joint. Because a complex number is a pair of real numbers, its probability law is defined by the joint distribution of the real and imaginary parts: for a vector \( X = X_R + jX_I \), the probability space is the joint distribution of \( X_R \) and \( X_I \)[1](https://tensorlab.cms.caltech.edu/users/anima/teaching/introduction-to-probability.pdf). Complex random vectors are the standard description of baseband noise and signals in digital communications, radar, and array processing, where demodulated waveforms are naturally complex[2](https://ocw.mit.edu/courses/6-450-principles-of-digital-communications-i-fall-2006/49163236e20779bae41639ff9dec1ac4_book_7.pdf).
| Key fact | Detail |
|---|---|
| Definition | An n-dimensional complex random vector \( Z = (Z_1, \ldots, Z_n)^T \) is specified by the joint distribution of its \( 2n \) real and imaginary components[2](https://ocw.mit.edu/courses/6-450-principles-of-digital-communications-i-fall-2006/49163236e20779bae41639ff9dec1ac4_book_7.pdf) |
| Second-order description | Complete second-order characterization requires the covariance matrix \( K_Z = E[ZZ^{\dagger}] \) and the pseudo-covariance matrix \( M_Z = E[ZZ^T] \)[2](https://ocw.mit.edu/courses/6-450-principles-of-digital-communications-i-fall-2006/49163236e20779bae41639ff9dec1ac4_book_7.pdf) |
| Properness | A complex random vector is proper (also called symmetrical or circular in the Gaussian case) when its pseudo-covariance vanishes[3](https://ieeexplore.ieee.org/document/243446/) |
| Affine invariance | Properness is preserved under affine transformations \( AZ + b \)[3](https://ieeexplore.ieee.org/document/243446/) • [1](https://tensorlab.cms.caltech.edu/users/anima/teaching/introduction-to-probability.pdf) |
| Maximum entropy | Among complex random vectors with a fixed correlation matrix, differential entropy is maximum if and only if the vector is proper, Gaussian, and zero-mean[3](https://ieeexplore.ieee.org/document/243446/) |
| Standard notation | A circularly-symmetric Gaussian variable is written \( Z \sim \mathcal{CN}(0, \sigma^2) \), with real and imaginary parts independent, each of variance \( \sigma^2/2 \)[2](https://ocw.mit.edu/courses/6-450-principles-of-digital-communications-i-fall-2006/49163236e20779bae41639ff9dec1ac4_book_7.pdf) |
## Second-order statistics: covariance and pseudo-covariance
For a real random vector, the covariance matrix captures all second moments. For a complex random vector, one second-central-moment matrix is not enough. The usual covariance matrix uses the conjugate transpose,
\[ K_Z = E[ZZ^{\dagger}], \]
while the pseudo-covariance matrix (also called the complementary covariance matrix or relation matrix) replaces the Hermitian transpose with an ordinary transpose,
\[ M_Z = E[ZZ^T]. \]
The pseudo-covariance is symmetric rather than Hermitian[2](https://ocw.mit.edu/courses/6-450-principles-of-digital-communications-i-fall-2006/49163236e20779bae41639ff9dec1ac4_book_7.pdf). The pair \( (K_Z, M_Z) \) must satisfy joint conditions to be realizable as the covariance and complementary covariance of some complex random vector, and the complementary matrix plays a role in power and entropy calculations[4](https://www.cambridge.org/core/books/abs/statistical-signal-processing-of-complexvalued-data/introduction-to-complex-random-vectors-and-processes/D8BAC4506CAF8BE96BEBB9631AC552F2).
In terms of the real and imaginary parts, properness means \( \mathrm{Cov}(XX^T) = 0 \), which is equivalent to the equal-variance and skew-correlation conditions \( \mathrm{Cov}(X_R, X_R^t) = \mathrm{Cov}(X_I, X_I^t) \) and \( \mathrm{Cov}(X_R, X_I^t) = -\mathrm{Cov}(X_I, X_R^t) \)[1](https://tensorlab.cms.caltech.edu/users/anima/teaching/introduction-to-probability.pdf). When a vector is proper, all of its second-order statistics are contained in the Hermitian covariance alone, and complex algebra can be carried out at a lower dimension by replacing transposes with Hermitian transposes[1](https://tensorlab.cms.caltech.edu/users/anima/teaching/introduction-to-probability.pdf).
## Properness and circular symmetry
Neeser and Massey, in their 1993 paper in the IEEE Transactions on Information Theory, defined complex random variables and processes with a vanishing pseudo-covariance as proper, and showed that the covariance of a complex random quantity, defined consistently with the real case, is determined by the usual complex covariance together with the pseudo-covariance[3](https://ieeexplore.ieee.org/document/243446/). They proved that properness is preserved under affine transformations and that the complex-multivariate Gaussian density takes its natural form only for proper random vectors[3](https://ieeexplore.ieee.org/document/243446/).
Circular symmetry is the distribution-level counterpart. A complex random vector is circularly symmetric if its distribution is the same as that of \( e^{i\theta}Z \) for every phase angle \( \theta \)[2](https://ocw.mit.edu/courses/6-450-principles-of-digital-communications-i-fall-2006/49163236e20779bae41639ff9dec1ac4_book_7.pdf). For a zero-mean, jointly Gaussian complex vector, circular symmetry holds if and only if the pseudo-covariance matrix is zero, so for Gaussian vectors properness and circular symmetry coincide[2](https://ocw.mit.edu/courses/6-450-principles-of-digital-communications-i-fall-2006/49163236e20779bae41639ff9dec1ac4_book_7.pdf). A proper [complex random variable](https://www.edgepedia.org/complex-random-variable) is also described as uncorrelated with its own complex conjugate[5](https://doi.org/10.1109/TSP.2011.2162954).
## Complex Gaussian vectors
A complex random vector is jointly Gaussian when its \( 2n \) real and imaginary components are jointly Gaussian; it is circularly-symmetric Gaussian when it is jointly Gaussian and circularly symmetric[2](https://ocw.mit.edu/courses/6-450-principles-of-digital-communications-i-fall-2006/49163236e20779bae41639ff9dec1ac4_book_7.pdf). For a single circularly-symmetric Gaussian variable \( Z \sim \mathcal{CN}(0, \sigma^2) \), the amplitude is Rayleigh distributed and the phase is uniform, and the real and imaginary parts are independent with variance \( \sigma^2/2 \) each[2](https://ocw.mit.edu/courses/6-450-principles-of-digital-communications-i-fall-2006/49163236e20779bae41639ff9dec1ac4_book_7.pdf).
For a proper complex Gaussian vector with mean \( m \) and nonsingular covariance matrix \( \Lambda \), the probability density is completely specified by the mean vector and the conventional covariance matrix, with no separate pseudo-covariance term[6](https://www.isiweb.ee.ethz.ch/papers/arch/fneese-mass-inspec-1993-1.pdf). Two jointly proper Gaussian vectors are independent if and only if their cross-covariance matrix vanishes, because uncorrelatedness and independence are equivalent for Gaussian variables[6](https://www.isiweb.ee.ethz.ch/papers/arch/fneese-mass-inspec-1993-1.pdf).
The maximum-entropy theorem generalizes to this setting: for a complex, continuous, n-dimensional random vector with nonsingular correlation matrix, differential entropy is bounded above, with equality if and only if the vector is proper, Gaussian, and zero-mean[3](https://ieeexplore.ieee.org/document/243446/) • [6](https://www.isiweb.ee.ethz.ch/papers/arch/fneese-mass-inspec-1993-1.pdf). This result is used, for example, to compute the capacity of a channel with proper complex Gaussian noise[6](https://www.isiweb.ee.ethz.ch/papers/arch/fneese-mass-inspec-1993-1.pdf).
## Improper vectors and widely linear processing
Vectors with a nonzero pseudo-covariance are called improper, or second-order noncircular. Many real signals are of this kind, and the improper nature of most complex-valued signals is the reason widely linear processing has become a standard approach[7](https://www.sciencedirect.com/science/article/abs/pii/S0165168412000813). A widely linear (also called linear-conjugate-linear) transformation acts on both the vector and its conjugate, which is what linear operations on the real and imaginary parts become when written in complex form[4](https://www.cambridge.org/core/books/abs/statistical-signal-processing-of-complexvalued-data/introduction-to-complex-random-vectors-and-processes/D8BAC4506CAF8BE96BEBB9631AC552F2).
The practical consequence is in estimation. For a strictly linear minimum mean-squared-error estimator to reach optimal performance, proper joint Gaussianity is sufficient, though not necessary; for improper signals, the widely linear estimator proposed by Bernard Picinbono and Pascal Chevalier in 1995, which processes the signal and its conjugate jointly, achieves a lower mean-squared error[8](https://www.sciencedirect.com/science/article/pii/S0165168419301951) • [13](https://www.probabilitycourse.com/chapter9/9_1_7_estimation_for_random_vectors.php). Widely linear methods have since been applied across estimation, filtering, detection, and equalization, and appear in modern designs such as widely linear precoding for improper signals in multi-user MISO systems, where the transmitted signal takes the form \( x = P_1 s + P_2 s^* \)[7](https://www.sciencedirect.com/science/article/abs/pii/S0165168412000813) • [9](https://www.nature.com/articles/s41598-026-53869-7).
Because propriety is often assumed implicitly, testing for it is a recognized problem: a complex-valued vector random sequence is proper when it is uncorrelated with its complex conjugate, and statistical tests exist for stationary vector sequences, including non-Gaussian ones[10](https://doi.org/10.1109/tsp.2017.2679681).
## Applications
Complex random vectors arise wherever two real channels are naturally paired. Noise waveforms after demodulation to baseband are complex and are best regarded as complex random vectors[2](https://ocw.mit.edu/courses/6-450-principles-of-digital-communications-i-fall-2006/49163236e20779bae41639ff9dec1ac4_book_7.pdf). Modulation schemes such as M-QAM, QPSK, and 8-PSK produce complex-valued signals, and radar, sensor array processing, spectral analysis of time series, and magnetic resonance imaging all lead to inherently complex-valued data[11](https://aaltodoc.aalto.fi/server/api/core/bitstreams/0a3aa11b-f265-4b44-875d-f2b01fcd45b4/content). In MIMO communication systems, the elements of the channel matrix and the noise vector are routinely modeled as complex Gaussian random quantities, with covariance matrices central to linear estimation and detection[12](https://www.frontiersin.org/journals/communications-and-networks/articles/10.3389/frcmn.2023.968370/full).
For proper complex random processes, Neeser and Massey derived a discrete [Fourier transform](https://www.edgepedia.org/fourier-transform) correspondence relating circular stationarity in the time domain to uncorrelatedness in the frequency domain, an information-theoretic application of the theory[3](https://ieeexplore.ieee.org/document/243446/).
## References
1. [Introduction to Probability lecture notes, Caltech (Complex Random Vectors)](https://tensorlab.cms.caltech.edu/users/anima/teaching/introduction-to-probability.pdf)
2. [Robert Gallager, Chapter 7, Random Processes and Noise, MIT 6.450 Principles of Digital Communications I, Fall 2006](https://ocw.mit.edu/courses/6-450-principles-of-digital-communications-i-fall-2006/49163236e20779bae41639ff9dec1ac4_book_7.pdf)
3. [F. D. Neeser and J. L. Massey (1993). Proper complex random processes with applications to information theory. IEEE Transactions on Information Theory 39(4), 1293–1302.](https://ieeexplore.ieee.org/document/243446/)
4. [Schreier and Scharf, Statistical Signal Processing of Complex-Valued Data, Chapter 2, Cambridge University Press (2010)](https://www.cambridge.org/core/books/abs/statistical-signal-processing-of-complexvalued-data/introduction-to-complex-random-vectors-and-processes/D8BAC4506CAF8BE96BEBB9631AC552F2)
5. [Complex-Valued Signal Processing: The Proper Way to Deal With Impropriety, IEEE Transactions on Signal Processing (2011)](https://doi.org/10.1109/TSP.2011.2162954)
6. [Neeser and Massey (1993), paper excerpt, ETH Zurich](https://www.isiweb.ee.ethz.ch/papers/arch/fneese-mass-inspec-1993-1.pdf)
7. [Navarro-Moreno et al. (2012). Linear and nonlinear filters based on the improper Karhunen–Loève expansion. Signal Processing 92(10).](https://www.sciencedirect.com/science/article/abs/pii/S0165168412000813)
8. [Measuring impropriety in complex and real representations, Signal Processing (2019)](https://www.sciencedirect.com/science/article/pii/S0165168419301951)
9. [Widely linear precoding with interference exploitation for improper signals in MU-MISO systems, Scientific Reports (2026)](https://www.nature.com/articles/s41598-026-53869-7)
10. [On Testing for Impropriety of Multivariate Complex-Valued Random Sequences, IEEE Transactions on Signal Processing (2017)](https://doi.org/10.1109/tsp.2017.2679681)
11. [Complex-valued signal processing review, Aalto University](https://aaltodoc.aalto.fi/server/api/core/bitstreams/0a3aa11b-f265-4b44-875d-f2b01fcd45b4/content)
12. [A review on principles, performance and complexity of linear estimation and detection techniques for MIMO systems, Frontiers in Communications and Networks (2023)](https://www.frontiersin.org/journals/communications-and-networks/articles/10.3389/frcmn.2023.968370/full)
13. [9 1 7 Estimation for Random Vectors, probabilitycourse.com](https://www.probabilitycourse.com/chapter9/9_1_7_estimation_for_random_vectors.php)
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*Topic: Encyclopedia › Physical world and mathematics › Mathematics and statistics › Statistics and probability › Probability theory › Random variables › Algebra and transformations of random variables › Complex random variables and processes-as-variables*
*Initially written Sep 17, 2026 · Reviewed: Sep 30, 2026; Oct 15, 2026 · Edited: Sep 30, 2026; Oct 15, 2026 · Last review: Oct 15, 2026*
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