Image Reject ratio derivation

[Element_115]

Does anyone know where I can find the derivation to these eqations:


IRR(phase)= 1 +4(cot(Δφ))²≈ 4/{(Δφ)²}

IRR(gain) ={(A+B)/(A-B)} = [{1+(1+ε)}/{1+(1+ε)}] ≈ 4/ε²

IRR(total) ≈ 4/{(Δφ)² + ε²}

These EQ's are in Thomas Lee's book "Design of RF CMOS IC"
2nd edition pg702

If anyone could explain how or refer me to some papers it would be greatly
appreciated.
Thanks in advance

 

[dazhen]

Equation 1

IRR(phase) = 1 +4(cot(Δφ))² = 1+4[cos(Δφ)/sin(Δφ)]² ≈ 1+4{[1-(Δφ)²/2]/Δφ}²
≈ 1+4[1-(Δφ)²]/(Δφ)² = 4/{(Δφ)²}-3 ≈ 4/{(Δφ)²}

Equation 2 seems not right. You may key in something wrong.

Apply 1/IRR(total) = 1/IRR(phase) + 1/IRR(gain), you can easily obtain equation 3.

 

[flatulent]

Try drawing a vector diagram and then with the error terms calculate the portion of the nominally orthogonal vectors that each project onto the other one.