• periodic signal: 1-D, 2-D

• special 2-D signal with 1-D-like periodicity and its Double Fourier Transform

• Kolmogorov's extension theorem

  • strict-sense stationary p: 1st-order, 2nd-order, Nth-order

  • strict-sense cyclostationary p: 1st-order, 2nd-order, Nth-order

• 2nd Moments

  • real-valued r.p.: WSS, covariance stationary, WSCS

  • proper-complex r.p.: WSS, covariance cyclostationary, WSCS

• (cont.)

  • periodic in translation variable

  • improper-complex SOS r.p.

  • improper-complex SOCS r.p.

• Two jointly distributed real-valued random processes

  • independent 

• (cont.)

  • uncorrelated, orthogonal,

  • jointly stationary in the strict sense, jointly WSS,

  • jointly cyclostationary with cycle period T in the strict sense, jointly WSCS with cycle period T

• Two jointly proper-complex random processes

  • uncorrelated

  • orthogonal (?)

  • jointly WSS

  • jointly WSCS with cycle period T

• Real-valued bandpass WSS random process

• (cont.)

  • mean and auto-correlation functions

  • complex envelope's mean, auto-correlation, and complementary auto-correlation functions

  • in-phase and quadrature-phase components' mean, auto-correlation, and cross-correlation functions

• Real-valued bandpass WSCS random process with cycle period T

  • mean and auto-correlation functions

  • complex envelope's mean, auto-correlation, and complementary auto-correlation functions

  • in-phase and quadrature-phase components' mean, auto-correlation, and cross-correlation functions

• Real-valued bandpass jointly WSS random processes

  • relation with jointly proper-complex jointly WSS random processes

• Real-valued bandpass jointly WSCS random processes

  • relation with jointly proper-complex jointly WSCS random processes

• Complex-valued random variable

  • real composite vector,

  • redundant complex augmented vector

  • pdf, CF; Gaussian random variable,

• (cont.)

  • circularity, propriety

• Complex-valued random vector

  • joint pdf, CF; Gaussian random vector,

  • circularly symmetric vs. proper vs. spherically symmetric

  • real composite vector, redundant complex augmented vector

• linear operation on x vs. linear operation on real composite vector. Are they the same?

• (cont.)

  • strictly linear operation on real composite vector of x = widely-linear (linear conjugate-linear) operation on x

  • linear operation on complex x = linear filtering of real bandpass signal

• real part operation

  • widely linear operation followed by real part operation = strictly linear operation followed by real part operation = widely linear operation without real part operation

• Inner product of real composite vector = real part of inner product of x = one-half the inner product of augmented vectors

• Quadratic form of real composite vector = quadratic of x with a Hermitian symmetric matrix + real part of quadratic~of x with a complex symmetric matrix = one-half the quadratic of augmented vector with a (Hermitian symmetric) matrix  = widely quadratic form

• Widely-Linear and Linear filtering of CT complex-valued signal

  • Widely-Linear Time-Varying (W-LTV) vs. LTV filtering

  • Widely-Linear Time-Invariant (W-LTI) vs. LTI filtering

  • Real-part operation

  • Inner product

• (cont.)

  • Widely Quadratic and Quadratic forms for CT complex signal

• Transform of proper-complex and improper-complex random vectors and 2nd-order statistics

  • Linear and Widely-Linear transforms

  • Inner product

  • Real-part operation

  • Quadratic form

• Examples:

  • L and WL transform of Gaussian random vector and CDF

  • real: SL

  • complex: proper, improper; SL, WL

  • Generation of a Gaussian random vector

    • real

    • complex: proper, improper

  • Conditional CDF of Gaussian random vector (Gallager p. 153) and L vs. WL transform

• Transform of proper-complex and improper-complex DT and CT random processes and 2nd-order statistics

• Properization?

• Improper-complex random variable and SS3.2 circularity coefficient

  • The correlation coefficient is not rotation invariant.

    • correlation coefficient and rotation

    • Case study: complex Gaussian

  • The circularity coefficient is rotation invariant.

    • Z value, circularity coefficient and rotation.

    • Case study: complex Gaussian

• Improper-complex Gaussian random variable and uncorrelating transformation

• Improper-complex random vector and SS3.2 Circularity coefficients

  • Z vector

  • Strong Uncorrelating Transform (SUT)

• Def. Complex correlation/coherence coefficient between X and X^* vs. circularity/impropriety coefficient

  • (mu, sigma squared, sigma tilde squared) vs. (mu, sigma squared, rho)

  • relation between rho and k

  • magnitude of rho and range of k

  • invariance of k under strongly linear/affine transformation

  • uncorrelating transformation of X

  • generation of improper-complex Gaussian random variable

• Improper-complex random vector and SS3.2 Circularity coefficients

  • Strong Uncorrelating Transform (SUT)

    • Z vector and coherence matrix

    • SVD and Takagi factorization of complex symmetric matrix

    • SUT and Takagi factorization of coherence matrix

    • circularity/impropriety coefficient matrix

• • Generation of Improper-complex Gaussian random vector

    • Differential entropy of improper-complex Gaussian random vector

    • Proper part and K part

  • Invariance of K under invertible strongly linear/affine transform

  • Degree of Impropriety

    • maximally improper random vector

• Trouble in applying results from linear algebra

  • Widely Unitary Transform

  • augmented EVD

  • PCA of improper-complex random vector and rank reduction

  • Widely KL expansion of CT complex random process

    • Whitening of a CT proper-complex random process

    • SUT of a CT improper-complex random process

  • What if an invertible WL transform properizes X?

9-1 Complex Baseband Representation of Real-Valued CT Bandpass Random Process -> PAPR, PMEPR

• nonlinear PA, OBO, TWTA

• Crest factor

• observation interval, average power, peak value, CCDF

• proper-complex WSS Gaussian Wrms T, double exponential

4/18

• Multi-stream decomposition

  • OFDM with CP, UW, ZP,

  • Block transmission with CP, UW, ZP

  • DFT-S-OFDM with CP

• Two-variable functions

  • impulse response of an LTV system

  • impulse response of an LPTV system

  • auto-correlation and complementary auto-correlation functions of a cyclostationary random process

  • cross-correlation functions of filtered random processes

• Bi-Frequency Spectrum and Complementary Bi-Frequency Spectrum

  • Definitions of bi-frequency and complementary bi-frequency spectra

  • Derivation of bi-frequency spectrum for an SOCS random process

    • Relation with PSD, bandlimited case

  • Derivation of complementary bi-frequency spectrum for an SOCS random process

  • SOS case

• LPTV system

  • as modulator and filter banks, and

  • as filter and modulator banks

• Vector random process

  • mean, auto-correlation, complementary auto-correlation functions

  • proper, improper,

  • WSS, SOS, WSCS, SOCS

  • pairwise jointly WSS, SOS, WSCS, SOCS

  • PSD matrix, complementary PSD matrix

  • bi-frequency spectrum matrix, complementary bi-frequency spectrum matrix

• LTI filtering and Ryz, ~Ryz, Syx, ~Syz

  • correlation functions and bi-frequency spectra

  • complementary correlation functions and complementary bi-frequency spectra

  • band-limited random process and its bi-frequency spectra

• Modulated random process and its Rxx, ~Rxx, Sxx, ~Sxx

• Jointly WSS random processes and their bi-frequency spectra

• FRESH vectorizer and scalarizer

  • conversion of a SOCS random process into a vector-valued SOS band-limited random process

    • (complementary) autocorrelation matrix, (complementary) PSD matrix

• FRESH vectorizer

  • reference bandwidth, reference rate, reference pair

  • input bandwidth, Nyquist zone, number of Nyquist zones

  • block diagram, invertibility

• FRESH scalarizer

  • reference bandwidth, reference rate, reference pair

  • output bandwidth

  • block diagram, invertibility

• Results

  • Conversion of a scalar-valued SOCS r.p. into a vector-valued SOS r.p.

  • slicing irrelevance for proper WSCS

  • LTI filtering of x(t)

  • LPTV filtering of x(t)

• • LPTV filtering of x(t) (cont.)

  • Vectorized Fourier Transform and FRESH vectorizer

  • LTI filtering, LPTV filtering of SOCS X(t)

  • WL-TI filtering, WL-PTV filtering of x(t),

• review of FRESH vectorization

  • bi-frequency spectrum and complementary bi-frequency spectrum

  • zero-mean proper-complex SOCS r.p -> zero-mean proper-complex SOS r.p.

  • nonzero-mean proper-complex SOCS r.p. -> nonzero-mean proper-complex SOS r.p.

  • improper-complex SOCS r.p -> improper-complex SOS r.p.

• p-FRESH vectorization of zero-mean improper-complex SOS r.p.

  • half-Nyquist zones and no-impulse at the end-points of the zones

  • flip and conjugation of negative frequency part

  • p-FRESH scalarization: inverse operation

• p-FRESH vectorizer and scalarizer of zero-mean improper-complx SOCS r.p.

  • reference bandwidth, reference rate, reference pair

  • Conversion of

    • FRESH vectorization of a zero-mean improper-complex SOCS X(t) to have an improper-complex SOS r.p.

• Results

  • LTI filtering, LPTV filtering, SOS X(t), SOCS X(t)

• FRESH-properzier: CT and DT

  • CT, DT

  • Conversion of a scalar-valued improper-complex SOCS r.p. with cycle period T into a scalar-valued proper-complex SOCS r.p. with cycle period 2T

  • Conversion of a scalar-valued improper-complex SOCS r.p. with cycle period M into a scalar-valued proper-complex SOCS r.p. with cycle period 2M

  • LTI filtering, LPTV filtering, WL-TI filtering, WL-PTV filtering of x(t), SOS X(t), SOCS X(t)

• Shift-Invariant Subspace

  • number of generators, amount of shift

  • relation with FRESH vectorizer

  • Nyquist criterion: folded spectrum and VFT

  • Generalized Nyquist criterion and

  • Fundamental theorem

  • FT

•  

  • orthogonality vs. bi-orthogonality

  • total interference

• joint Tx and Rx optimization in cyclostationary noise

• cognitive radio

• orthogonal overlay

• rate reduction technique

• capacity

• orthogonal projection matrix

• constrained set partitioning

• cyclic water filling

• FDMA, TDMA, short-code CDMA

• LMMSE vs. MSINR

• Continuous-time equivalents of Welch bound equality sequences

•  FDMA vs. CDMA

• CT equivalents of Generalized WBE sequences

• Overloaded system and FTN

• sampling of a CT improper-complex SOCS random process

• block processing of a DT improper-complex SOCS random process

• pre-processing

• centered DFT

• Asymptotic FRESH properizer

• data-like co-channel interference
• block processing and asymptotic block matrix with diagonal blocks

• data-like improper-complex CCI
• block processing of augmented vector

• (COM) Symbol Detection: CT observation, DT observation, vector observation,

  • QAM over flat AWGN 

    • Tx: auto-correlation, PSD, bi-frequency spectrum, complementary bi-frequency spectrum

    • Rx: Nyquist Criterion for orthogonality, Generalized Nyquist Criterion, Bi-orthogonality

  • PAM over flat AWGN 

    • Tx: auto-correlation, PSD,bi-frequency spectrum, complementary bi-frequency spectrum

    • Rx: Nyquist Criterion for orthogonality, Generalized Nyquist Criterion, Bi-orthogonality

  • QAM over selective WSS, WSCS, SOS, SOCS noise

    • Rx: MF, LMMSE, LZF, DFE, WSS noise, WSCS noise

  • PAM over selective WSS, WSCS, SOS, SOCS noise

    • Rx: MF followed by real-part operation, LCL filtering, LZF, DFE, WSS noise, WSCS noise

• (COM) Orthogonal Overlay: rate-reduction technique

  • proper WSCS: MMSE

  • improper SOCS: MMSE

• (COM) Multiple-Access: CDMA, TDMA, FDMA, NOMA

  • Interference Function for Long-code CDMA

  • Welch-bound equality, GWBE, CTE-WBE sequences for short-code CDMA

• (COM) Block transmission with CP, UW, ZP in data-like interference

  • WSCS, SOCS

• (COM) DFT-spread OFDM with CP, UW, ZP

  • pi/2-BPSK, QPSK, OQPSK, PAM, QAM

  • GNC, PAPR

• (COM) FBMC

  • OQAM, QAM

   

• (Inf) Capacity of cyclostationary noise channel

  • proper-complex WSCS Gaussian noise

  • improper-complex SOCS Gaussian noise

• (Inf) Capacity of orthogonal overlay channel

  • proper WSCS

  • improper SOCS

• (Inf) Sum capacity of CDMA

  • single-code: D vs. CT

  • multi-code: D vs. CT

• (Inf) Capacity of sampled (FTN, Nyquist, sub-Nyquist)

  • proper WSS, WSCS Gaussian noise channel

  • improper SOS, SOCS Gaussian noise channel

• (Inf) Rate-distortion function of sampled (FTN, Nyquist, sub-Nyquist)

  • proper-complex WSS, WSCS Gaussian source

  • improper-complex SOS, SOCS Gaussian source

• mechanical and civil engineering: vibration and cyclostationarity

• biomedical engineering: speech signal, cardiogram

(SP) Sampling: band-limited, unlimited

• Uniform sampling of a CT SOS random process

• Uniform sampling of a CT SOCS random process

• Uniform Sampling:

  • FTN, Nyquist, sub-Nyquist

  • Sampling system: single-branch, multi-branch, modulation and filter banks

  • Prefilter: LTI and WL-TI

• Asymptotic FRESH-properizer

  • DT vs. discrete

  • DTFT vs. DFT

    • Complex envelope of real-valued bandpass WSS and WSCS random processes

    • Widely-Linear Periodically Time-Varying (W-LPTV) vs. LPTV filtering of a CT complex-valued signal

    • IQ imbalance compensation

    • Gardner's book

    two uncorrelated improper-complex random processes

  • two orthogonal real-valued random processes

  • two jointly proper-complex WSS random processes

• (SP) asymptotic properizer for D

• (SP) Estimation: LMMSE, LCLMMSE, FRESH filtering, p-FRESH filtering

• (SP) Detection: Presence detection: proper SOCS, improper SOCS,