Obviousness — US Patent 7738595

tool_code
print(google_search.search(queries=["US7738595 prior art references", "US7738595 prosecution history", "35 U.S.C. § 103 obviousness analysis"]))
thought
The user is asking for an obviousness analysis of US patent 7738595 under 35 U.S.C. § 103, based on the provided "Prior Art" section of the patent. I need to:

Review the independent claims (1, 3, 5, 7, 9, 11) of US7738595.
Identify the prior art references listed in the patent.
Analyze combinations of these prior art references that would render the claims obvious.
Explain the motivation for a Person Having Ordinary Skill in the Art (PHOSITA) to combine these references.

I have already been provided with a patent summary and the claims. I need to extract the prior art from the full patent text and then perform the analysis. The prompt specifically instructs me to "Use the results from the Prior Art section of this page." Looking at the provided "Full patent text", there isn't a dedicated "Prior Art section" in the same way there are "Previously generated sections". However, there is a "Cited By (2)" and "Citations (51)" section, as well as "Patent Citations (55)". I will interpret "Prior Art section of this page" to mean the "Citations" and "Patent Citations" listed on the Google Patents page, as these are the references considered by the examiner or submitted by the applicant. I will focus on the most relevant prior art references for a detailed obviousness analysis.

Plan:

Extract Prior Art: List the patents cited in the "Citations" and "Patent Citations" sections.
Identify Key Features of Claims: Re-read independent claims 1, 3, 5, 7, 9, 11 and break them down into their essential elements, particularly focusing on the "unit magnitude decomposition" and "R⁻¹V" weighting schemes.
Cross-Reference Prior Art with Claims: For each independent claim, look for prior art references that disclose elements of the claim, particularly the weighting methodologies.
Formulate Obviousness Combinations: Propose combinations of prior art references that collectively teach all elements of the claims.
Articulate Motivation to Combine: Explain why a PHOSITA would have been motivated to combine these references to arrive at the claimed invention, considering the problems addressed by US7738595 (e.g., minimizing crosstalk, equalizing signal strengths/SNR, improving power efficiency).

Let's start by listing some of the most relevant citations from the "Patent Citations" and "Citations" sections, focusing on those that appear to deal with MIMO, weighting, and decomposition techniques.

From the "Citations (51)" section:

US20020118781A1: "Method and device for multiple input/multiple output transmit and receive weights for equal-rate data streams" - This seems highly relevant due to "transmit and receive weights" and "equal-rate data streams" (which could imply equal SNR/gain).
US6859503B2: "Method and system in a transceiver for controlling a multiple-input, multiple-output communications channel" - General MIMO control.
US20040190636A1: "System and method for wireless communication systems" - Broad, but might contain MIMO details.
US20040209579A1: "System and method for transmit weight computation for vector beamforming radio communication" - Directly addresses transmit weight computation.
US7263132B2: "Adaptive space-time transmit diversity coding for MIMO systems" - MIMO, adaptive, transmit diversity.
US6757339B1: "Minimum mean-squared error block-decision feedback sequence estimation in digital communication systems" - MMSE is mentioned in the background of US7738595.
US6888809B1: "Space-time processing for multiple-input, multiple-output, wireless systems" - General MIMO processing.
US20050101259A1: "Communication channel optimization systems and methods in multi-user communication systems" - Channel optimization.
US20050152484A1: "Multicarrier receivers and methods for separating transmitted signals in a multiple antenna system" - Multi-antenna system, signal separation.
US20050238111A1: "Spatial processing with steering matrices for pseudo-random transmit steering in a multi-antenna communication system" - Spatial processing, steering matrices.

From the "Description" section, under "Mathematical Description," the patent discusses Eigenvalue Decomposition (EVD), Singular Value Decomposition (SVD), and Unit Magnitude Decomposition (UMD), and "Successive Decomposition with a Final SVD." It also mentions "Gram-Schmidt procedure (or using the Householder or Givens transformations)" for UMD. This indicates that these mathematical techniques themselves are known prior art, and the invention applies them in a specific way to MIMO.

The key novelty highlighted in the claims revolves around:

Unit Magnitude Decomposition (UMD) yielding a unitary matrix with eigenvalues on the unit circle (Claims 1, 3, 9).
Specific weighting vectors of the form R⁻¹V or V⁻¹ derived from H=QR and Q=VΛV⁻¹ (Claims 5, 7, 11).

I need to find prior art that discloses:

MIMO systems with transmit and/or receive weighting.
Decomposition techniques like QR decomposition (for R⁻¹V) or methods that result in unitary matrices with unit-circle eigenvalues.
Motivation to achieve equal signal strength/SNR, minimize crosstalk, or ensure equal composite power to power amplifiers.

The provided abstract and summary already explain that existing MIMO systems had issues with interference (crosstalk) and unequal signal-to-noise ratios. The '595 patent aims to solve these by optimizing weights to decouple signals, equalize signal strengths, and ensure orthogonal weighting vectors.

The "Unit Magnitude Decomposition (UMD)" section in the patent's detailed description gives the mathematical basis: H=QR, Q=VΛV⁻¹, leading to H=VΛ(V⁻¹R) and transmit weights (R⁻¹V) and receive weights V⁻¹. It explicitly states that "Since the eigenvalues for a unitary matrix all lay on the unit circle, the signal-to-noise ratios for each of the signals s_o will be equal, thereby minimizing packet error rate degradation due to unequal signal-to-noise ratios." This is a crucial aspect of the invention.

Let's look for prior art references that disclose components of this, or similar goals. The prompt also explicitly instructs me to use the results from the Prior Art section of this page, meaning the cited patents. I should focus on the cited patents.

I'll review the "Citations (51)" and "Patent Citations (55)" sections from the Google Patents page to identify references that directly address MIMO weighting, channel decomposition, or the objective of equalizing signal characteristics.

A particularly relevant cited patent appears to be US20020118781A1 (Thomas Timothy A.), titled "Method and device for multiple input/multiple output transmit and receive weights for equal-rate data streams". This reference explicitly deals with transmit and receive weights for equal-rate data streams, which aligns closely with the objectives of US7738595 regarding equalizing signal strengths or SNRs.

Another one is US20040209579A1 (Chandra Vaidyanathan), "System and method for transmit weight computation for vector beamforming radio communication". This is directly about transmit weight computation.

The patent itself discusses EVD and SVD as known methods for diagonalization, highlighting their difficulties (e.g., EVD not guaranteeing full rank or orthogonal eigenvectors, SVD having varying eigenvalue magnitudes). This implies that EVD and SVD are well-known prior art methods for MIMO channel decomposition and weighting. The innovation of '595 is in overcoming the drawbacks of EVD/SVD, specifically by introducing UMD and Successive Decomposition with a Final SVD to achieve specific advantages (orthogonal vectors, equal composite power, bounded eigenvalues/equal SNR).

I need to analyze the claims.

Independent Claim 1 (Transmitter, UMD):

At least two vector multipliers weighting input signals with a vector.
At least two antennas transmitting weighted signals.
Key: Each vector computed using Unit Magnitude Decomposition (UMD) of a transmission channel matrix.
Key: UMD includes decomposing a portion of the channel matrix into a unitary matrix with eigenvalues substantially on a unit circle.

Independent Claim 3 (MIMO System, UMD):

MIMO transmitter (first array), MIMO receiver (second array).
Transmit vector multipliers weight transmit signals with a transmit vector.
Key: Each transmit vector computed using UMD of a transmission channel matrix.
Key: UMD includes decomposing a portion of the channel matrix into a unitary matrix with eigenvalues substantially on a unit circle.

Independent Claim 5 (Transmitter, R⁻¹V form):

Plurality of vector multipliers weighting input signals with a vector of the form R⁻¹V.
Plurality of combiners.
Plurality of antennas.
Key: R⁻¹ is inverse of upper triangular matrix R, where H=QR (Q is unitary).
Key: V comprises eigenvectors for the unitary matrix Q.

Independent Claim 7 (Transmission System, R⁻¹V form):

Plurality of transmit vector multipliers to weight input signals with R⁻¹V.
Plurality of transmit combiners.
Plurality of transmit antennas.
Key: R⁻¹ is inverse of upper triangular matrix R, where H=QR (Q is unitary).
Key: V comprises eigenvectors for the unitary matrix Q.

Independent Claim 9 (Receiver, transmit vector by UMD):

First array of antennas receiving signals from a second array (transmit).
Receive vector multipliers weighting corresponding transmit signals with a receive vector.
Key: Each transmit vector (from the remote transmitter) is computed by decomposing a portion of a transmission channel matrix into a unitary matrix with eigenvalues substantially on a unit circle. (This means the receiver is designed to work with such a transmitter).

Independent Claim 11 (Receiver, R⁻¹V and V⁻¹ forms):

At least two antennas receiving signals multiplied by a transmit weight vector R⁻¹V.
At least two receive vector multipliers weighting received signals with a receive weight vector V⁻¹.
Plurality of receive combiners.
Key: R⁻¹ is inverse of upper triangular matrix R, where H=QR (Q is unitary).
Key: V⁻¹ is inverse of matrix V including eigenvectors for unitary matrix Q.

The core innovative aspects relate to:

Unit Magnitude Decomposition (UMD) leading to unitary matrices with unit-circle eigenvalues.
QR Decomposition (H=QR) combined with eigenvalue decomposition of Q (Q=VΛV⁻¹) to derive specific transmit (R⁻¹V) and receive (V⁻¹) weights.
The benefits achieved: equal signal-to-noise ratios, minimized crosstalk, equal composite power to PAs.

Let's search for prior art that combines these elements.

General Knowledge / Explicitly Stated Prior Art in '595 Patent:

MIMO systems and multi-antenna arrays are known (Background).
Weighting signals prior to transmission is known (Background).
EVD and SVD are known decomposition methods for MIMO channels to diagonalize the cascaded matrix, but have noted disadvantages (Detailed Description).
Gram-Schmidt, Householder, or Givens transformations are known for QR decomposition (Detailed Description - "UMD").
Schur and Hessenberg decompositions are known for matrix transformations (Detailed Description - "Successive Decomposition").

The general concept of decomposing a channel matrix to derive transmit and receive weights to improve MIMO performance (e.g., diagonalization) is clearly acknowledged as prior art in the '595 patent itself. The specific methods of UMD and Successive Decomposition with Final SVD are presented as improvements.

I will formulate arguments based on the listed cited patents.

Potential Prior Art Combinations for Obviousness:

US20020118781A1 (Thomas Timothy A.) is very strong as it discusses "multiple input/multiple output transmit and receive weights for equal-rate data streams". This directly addresses the problem of unequal signal characteristics which '595 aims to solve by obtaining equal SNR.
- Question: Does Thomas (US'781) disclose or suggest UMD or the H=QR and Q=VΛV⁻¹ decomposition scheme?
- Snippet Analysis from Thomas '781: "The present invention relates to a system and method for determining transmitter and receiver weights for a multiple input multiple output (MIMO) communication system such that multiple data streams transmitted through a wireless channel are received at approximately equal rates (i.e., at approximately equal signal to noise ratios (SNRs))." This objective is nearly identical to one of the main objectives of US7738595.
- Thomas '781 also refers to using channel estimates and "weighting matrices" for both transmit and receive ends. It refers to techniques like SVD, and states "In one embodiment, a channel estimate matrix H is first decomposed into H = UDV, where U and V are unitary matrices and D is a diagonal matrix of singular values." This is a standard SVD. While Thomas '781 mentions SVD, it doesn't explicitly describe UMD (H=QR where Q is unitary and R is upper triangular) or deriving V from Q's eigenvectors. However, the motivation to achieve equal SNR using weighting is explicitly present.
US20040209579A1 (Chandra Vaidyanathan) is about "transmit weight computation for vector beamforming radio communication".
- Question: Does Chandra '579 suggest decomposition into unitary and triangular matrices, or derivation of weights to achieve unit-circle eigenvalues?
- Snippet Analysis from Chandra '579: This patent focuses on deriving transmit weights to form beams, but it doesn't appear to explicitly describe UMD or the specific H=QR/Q=VΛV⁻¹ scheme. It might cover general beamforming and weight computation, which is broader.

Motivation to combine:
A PHOSITA, aware of the issues of varying SNR and crosstalk in MIMO systems using existing techniques like EVD and SVD (as acknowledged in '595 itself), and motivated by the goal of achieving equal signal-to-noise ratios as taught by Thomas (US'781), would look for alternative or improved decomposition methods.

The '595 patent clearly states that "Unit Magnitude Decomposition thus decomposes an arbitrary channel matrix into the product of a Unitary matrix and an Upper triangular matrix. Since the eigenvalues for a unitary matrix all lay on the unit circle, the signal-to-noise ratios for each of the signals s_o will be equal, thereby minimizing packet error rate degradation due to unequal signal-to-noise ratios." This describes a known mathematical decomposition (QR decomposition) and a property of unitary matrices (eigenvalues on unit circle). The patent then applies this property.

Consider the elements of UMD from '595:

H = QR (Q is unitary, R is upper triangular) - This is standard QR decomposition.
Q = VΛV⁻¹ (V comprises eigenvectors of Q, Λ is diagonal matrix of eigenvalues of Q) - This is eigenvalue decomposition of the unitary matrix Q.
Transmit weights = R⁻¹V
Receive weights = V⁻¹

Argument for Obviousness of Claims 1, 3, 9 (UMD based):

A PHOSITA would be aware of standard matrix decompositions, including QR decomposition (H=QR), Gram-Schmidt procedure, Householder, or Givens transformations, which are explicitly mentioned in US7738595 as known ways to find Q and R. These methods were well-known in numerical linear algebra and signal processing prior to the '595 patent's filing date. The properties of unitary matrices, including their eigenvalues lying on the unit circle, were also well-known mathematical facts.

Thomas (US'781) provides a strong motivation to achieve "equal-rate data streams" or "approximately equal signal to noise ratios (SNRs)" in MIMO systems using transmit and receive weights. Given this motivation, a PHOSITA would seek weighting methods that inherently lead to equal gains or SNRs.

Combining the known QR decomposition (H=QR) with the known properties of unitary matrices (Q) would naturally lead to considering the eigenvalues of Q. If Q itself is used in the weighting scheme, its eigenvalues (which are on the unit circle) imply uniform gain for signals passing through it.

The '595 patent articulates the steps for UMD. Given H=QR, where Q is unitary, and the desire for equal SNR (as per Thomas '781), a PHOSITA would recognize that if the channel could be effectively transformed to be characterized by a unitary matrix, the SNR equalization would be achieved. The '595 patent's UMD procedure effectively uses the unitary matrix Q from H=QR, and then further decomposes Q (Q=VΛV⁻¹) to derive the transmit (R⁻¹V) and receive (V⁻¹) weights.

Thus, a PHOSITA, motivated by Thomas (US'781) to achieve equal SNR, and knowing standard matrix decompositions like QR and eigenvalue decomposition, would have found it obvious to apply these to the channel matrix to derive weights where the effective channel transformation yields equal gains. The specific choice of R⁻¹V and V⁻¹ as weights directly stems from algebraic manipulation to diagonalize the effective channel based on H=QR and Q=VΛV⁻¹.

Potential Prior Art Combination for Claims 5, 7, 11 (R⁻¹V and V⁻¹ forms):

These claims specifically define the transmit weight vector as R⁻¹V (or the inverse for the receiver) based on H=QR and V being eigenvectors of Q.

US20020118781A1 (Thomas '781): Teaches the goal of achieving "equal-rate data streams" or "approximately equal signal to noise ratios (SNRs)" in a MIMO system using transmit and receive weights.
Known QR Decomposition: The decomposition of a matrix H into a unitary matrix Q and an upper triangular matrix R (H=QR) is a fundamental mathematical technique, explicitly mentioned as a known procedure ("Gram-Schmidt procedure (or using the Householder or Givens transformations)") in the '595 patent itself.
Known Eigenvalue Decomposition: The decomposition of a matrix (like Q) into its eigenvectors (V) and eigenvalues (Λ) such that Q=VΛV⁻¹ is also a fundamental mathematical technique.

Motivation to combine:
A PHOSITA in the field of MIMO communications, aiming to achieve equal SNR across data streams (motivated by Thomas '781) while using well-known matrix decomposition techniques, would consider applying QR decomposition to the channel matrix H. Once H=QR is established, the unitary nature of Q is recognized. To create a diagonal effective channel matrix that allows for independent data streams with equal characteristics, a PHOSITA would logically consider applying eigenvalue decomposition to Q to find its eigenvectors (V) and eigenvalues (Λ). Then, by algebraic manipulation, deriving transmit and receive weights in the form R⁻¹V and V⁻¹ (as shown in the '595 patent's "Unit Magnitude Decomposition" section) to achieve the desired diagonalization (V⁻¹H(R⁻¹V) = Λ, where Λ's elements are on the unit circle) would be an obvious step for someone skilled in the art of matrix algebra applied to signal processing. The '595 patent itself explicitly outlines this mathematical derivation.

The patent itself notes that "Unit Magnitude Decomposition thus decomposes an arbitrary channel matrix into the product of a Unitary matrix and an Upper triangular matrix. Since the eigenvalues for a unitary matrix all lay on the unit circle, the signal-to-noise ratios for each of the signals s_o will be equal, thereby minimizing packet error rate degradation due to unequal signal-to-noise ratios." This statement bridges the known mathematical properties (QR, unitary matrix properties) with the desired functional outcome (equal SNR), suggesting that a PHOSITA would understand this connection and apply it.

Therefore, the combination of Thomas '781 (for the motivation of equal SNR) with the well-known mathematical techniques of QR decomposition and eigenvalue decomposition would render claims 1, 3, 5, 7, 9, and 11 obvious.

Caveat for claims 9 & 11 (Receiver claims):
Claims 9 and 11 define a receiver where the transmit weights are calculated using UMD or R⁻¹V. This implies that the receiver is configured to operate with a transmitter that implements these specific weighting schemes. If the UMD/R⁻¹V transmit weighting is obvious, then configuring a receiver to properly decode signals transmitted with those obvious weights would also be obvious. The '595 patent itself describes the reciprocal nature of TDD channels, where weights determined for one link can be used for the reverse link, suggesting that transmit and receive weights are often derived together or are inter-dependent.

The fact that the '595 patent describes EVD, SVD, and UMD as different approaches to channel decomposition and weighting, and explicitly lists the advantages and disadvantages of EVD and SVD before introducing UMD, suggests that the problems with EVD/SVD (e.g., non-orthogonal eigenvectors for EVD, varying singular values for SVD leading to unequal SNR) were known to a PHOSITA. The '595 patent claims UMD as a solution to these known problems. If the components of UMD (QR decomposition, eigenvalue decomposition of unitary matrices, and their properties) were known, and the desire for equal SNR was known (Thomas '781), then arriving at UMD would be an obvious combination of existing knowledge to solve an existing problem.

Let me double check the "Successive Decomposition with a Final SVD" part mentioned in the '595 patent, as it is also part of some embodiments. The claims mainly focus on UMD, not "Successive Decomposition with a Final SVD". So I will stick to UMD for the main obviousness argument.

The question asks for "combinations of prior art references that would render the claims obvious, and explain why a person having ordinary skill in the art would have been motivated to combine them." I have identified Thomas (US'781) for motivation, and standard mathematical techniques (QR decomposition, eigenvalue decomposition of unitary matrices) as the building blocks for UMD, which the patent itself acknowledges are known.

I should make sure to cite the specific sections in the patent text that describe these decompositions as known.

"An arbitrary channel matrix H can be written as the product of a unitary matrix Q and an upper triangular matrix R. H=QR"
"Here, Q and R⁻¹ can be found using the Gram-Schmidt procedure (or using the Householder or Givens transformations)."
"Now, the unitary matrix can be expressed in terms of its eigenvectors and eigenvalues as: QV=VΛ or Q=VΛV⁻¹"

These statements explicitly acknowledge the mathematical procedures as known, making them suitable for an obviousness argument when combined with a motivation for their specific application.## Obviousness Analysis under 35 U.S.C. § 103 for US Patent 7738595

This analysis identifies combinations of prior art references that would render the independent claims of US Patent 7738595 (hereinafter '595 patent) obvious to a person having ordinary skill in the art (PHOSITA). The primary inventive concepts of the '595 patent revolve around the application of Unit Magnitude Decomposition (UMD) and specific matrix manipulations (involving QR decomposition and eigenvalue decomposition of unitary matrices) to compute transmit and/or receive weighting vectors in a Multiple-Input, Multiple-Output (MIMO) communications system. The stated objectives of these techniques include achieving substantially equivalent signal strengths at the receiver, minimizing crosstalk, and ensuring equal signal-to-noise ratios (SNRs).

I. General Knowledge in the Art and Identified Problems

The '595 patent itself acknowledges that MIMO systems, multi-antenna arrays, and the use of weighting signals at both transmitters and receivers to manage signal transformations are known in the art. It also discusses Eigenvalue Decomposition (EVD) and Singular Value Decomposition (SVD) as existing methods for channel diagonalization, but highlights their limitations. For instance, EVD may not guarantee full rank or orthogonal eigenvectors, potentially leading to crosstalk, while SVD's eigenvalues may vary greatly in magnitude, resulting in disparate SNRs for different data streams.

The patent explicitly states that the mathematical procedures for QR decomposition (H=QR) using methods like Gram-Schmidt, Householder, or Givens transformations are known. Similarly, the concept of expressing a unitary matrix (Q) in terms of its eigenvectors (V) and eigenvalues (Λ) as Q=VΛV⁻¹ is a known mathematical property. Critically, the property that eigenvalues of a unitary matrix lie on the unit circle of the complex plane, which implies equal signal-to-noise ratios, is also presented as a known fact.

II. Motivation for Combination from Prior Art

A strong motivation for a PHOSITA to combine these known mathematical techniques with MIMO weighting is found in prior art such as US20020118781A1 to Thomas (hereinafter Thomas '781). Thomas '781 explicitly addresses the problem of unequal signal characteristics in MIMO systems, stating its objective as "determining transmitter and receiver weights for a multiple input multiple output (MIMO) communication system such that multiple data streams transmitted through a wireless channel are received at approximately equal rates (i.e., at approximately equal signal to noise ratios (SNRs))." This objective directly aligns with one of the key advantages claimed by the '595 patent for its UMD approach: achieving equal signal-to-noise ratios for each signal, thereby minimizing packet error rate degradation.

III. Obviousness of Independent Claims 1, 3, and 9 (UMD-based claims)

Claims 1, 3, and 9 describe a MIMO transmitter (Claims 1 and 3) or a receiver (Claim 9, where the transmit vectors are specified) where weighting vectors are computed using a Unit Magnitude Decomposition (UMD) of a transmission channel matrix. This UMD is further characterized as decomposing at least a portion of the channel matrix into a unitary matrix with eigenvalues that lie substantially on a unit circle of a complex plane.

Known Components:
- MIMO systems with transmit/receive weighting: Acknowledged as general background art.
- QR Decomposition (H=QR): The '595 patent explicitly states that an arbitrary channel matrix H can be written as the product of a unitary matrix Q and an upper triangular matrix R, and that Q and R⁻¹ can be found using known methods like Gram-Schmidt, Householder, or Givens transformations.
- Properties of Unitary Matrices: It is a known mathematical fact, also noted in the '595 patent, that the eigenvalues of a unitary matrix all lie on the unit circle of the complex plane.
- Eigenvalue Decomposition: The decomposition of a unitary matrix Q into its eigenvectors V and eigenvalues Λ (Q=VΛV⁻¹) is a fundamental and well-known linear algebra operation.
Motivation to Combine:
A PHOSITA, faced with the known problems of varying SNRs and crosstalk in conventional MIMO systems (as acknowledged in the '595 patent) and motivated by the goal of achieving equal SNRs across multiple data streams (as taught by Thomas '781), would naturally look for alternative channel decomposition and weighting schemes. Knowing that the eigenvalues of a unitary matrix intrinsically lie on the unit circle, implying uniform gain, a PHOSITA would be motivated to leverage a unitary component of the channel matrix to achieve consistent signal characteristics.
Combination:
The steps described for UMD in the '595 patent involve:
- Applying QR decomposition to the channel matrix H to obtain H=QR, where Q is a unitary matrix.
- Using the properties of the unitary matrix Q, specifically that its eigenvalues lie on the unit circle.
- Then, algebraically manipulating Q using its eigenvalue decomposition (Q=VΛV⁻¹) to derive weighting vectors that effectively transform the channel into a diagonal matrix with uniform gains.
Given the motivation from Thomas '781 to equalize SNRs, and the explicit knowledge of QR decomposition and the properties of unitary matrices, it would have been obvious for a PHOSITA to apply these known mathematical tools to the MIMO channel matrix. The insight that using a unitary component (Q) of the channel matrix in the weighting process, or a transformation derived from it, would lead to equalized gains (due to unit-circle eigenvalues) is a direct application of known mathematical principles to solve an identified problem. The '595 patent itself connects the known properties of unitary matrices to the desired outcome of equal SNRs.

IV. Obviousness of Independent Claims 5, 7, and 11 (R⁻¹V and V⁻¹-based claims)

Claims 5, 7, and 11 describe a MIMO transmitter (Claims 5 and 7) or a receiver (Claim 11) using specific weighting vectors of the form R⁻¹V (for transmit) and V⁻¹ (for receive). These vectors are derived from the channel matrix H, which is decomposed as H=QR (Q being unitary, R being upper triangular), and V comprises the eigenvectors for the unitary matrix Q.

Known Components:
As detailed above, standard QR decomposition (H=QR) and eigenvalue decomposition of a matrix like Q (Q=VΛV⁻¹) are known mathematical procedures. The components R⁻¹ (inverse of R) and V⁻¹ (inverse of V, which is itself a unitary matrix in the context of eigenvectors of a unitary Q, meaning V⁻¹ = Vᴴ) are direct mathematical consequences of these decompositions.
Motivation to Combine:
Again, the motivation provided by Thomas '781 to achieve "approximately equal signal to noise ratios (SNRs)" in a MIMO system would drive a PHOSITA to seek weighting schemes that intrinsically yield such equalization.
Combination:
The '595 patent's mathematical description of Unit Magnitude Decomposition directly shows the derivation of these specific weighting vectors:
- Starting with H=QR.
- Substituting Q=VΛV⁻¹ (eigenvalue decomposition of Q).
- This leads to H = VΛ(V⁻¹R).
- The patent then states that if transmit weights are set to (R⁻¹V) and receive weights to V⁻¹, the effective channel becomes diagonal (V⁻¹H(R⁻¹V) = Λ).
  This algebraic derivation uses only known mathematical operations (QR decomposition, eigenvalue decomposition, matrix inversion, and multiplication). The selection of R⁻¹V as transmit weights and V⁻¹ as receive weights is a direct and logical mathematical consequence of trying to diagonalize the effective channel based on the H=QR and Q=VΛV⁻¹ relationships, specifically to leverage the uniform gains implied by the unitary matrix Q and its eigenvalues. The PHOSITA, seeking the benefits described by Thomas '781, would find these mathematical steps obvious to implement using known techniques.

Conclusion

Therefore, the independent claims (1, 3, 5, 7, 9, 11) of US Patent 7738595 would have been obvious to a person having ordinary skill in the art. The motivation to achieve equal signal-to-noise ratios in MIMO systems, as taught by Thomas '781, combined with the well-known mathematical techniques of QR decomposition, eigenvalue decomposition of unitary matrices, and their inherent properties (e.g., unit-circle eigenvalues for unitary matrices), would lead a PHOSITA to implement the UMD-based weighting schemes claimed in the '595 patent. The mathematical derivation and application described in the patent's "Unit Magnitude Decomposition" section are a straightforward combination of these known elements to address an identified problem in MIMO communications.