Common-mode and quadrature components of the signal

An example of how a phase-modulated signal ( green line ) decomposes into two components: in-phase

I(t)

{\ displaystyle I (t)}

I (t)

and quadrature

Q(t)

{\ displaystyle Q (t)}

Q (t)

.

Common-mode and quadrature components - the result of the analog signal $S(t)$ ${\ displaystyle S (t)}$ $S (t)$ in the form of a combination of real and imaginary parts ^[1] :

S(t)=A_{1}(t)\cos(\omega t)+A_{2}(t)\sin(\omega t)

{\ displaystyle S (t) = A_ {1} (t) \ cos (\ omega t) + A_ {2} (t) \ sin (\ omega t)}

{\ displaystyle S (t) = A_ {1} (t) \ cos (\ omega t) + A_ {2} (t) \ sin (\ omega t)}

,

where the first term is called the in-phase component (or I-component , from the English in-phase ) of the signal $S(t)$ ${\ displaystyle S (t)}$ $S (t)$ , the second is called the quadrature component (or Q-component , from the English quadrature ) of the signal $S(t)$ ${\ displaystyle S (t)}$ $S (t)$ :

I(t)=A_{1}(t)\cos(\omega t)

{\ displaystyle I (t) = A_ {1} (t) \ cos (\ omega t)}

{\ displaystyle I (t) = A_ {1} (t) \ cos (\ omega t)}

Q(t)=A_{2}(t)\sin(\omega t)

{\ displaystyle Q (t) = A_ {2} (t) \ sin (\ omega t)}

{\ displaystyle Q (t) = A_ {2} (t) \ sin (\ omega t)}

Although this decomposition can be obtained for any signal, it is of most interest for narrow-band signals, that is, for signals with a small spectrum width. For such signals, $A_{1}(t)$ ${\ displaystyle A_ {1} (t)}$ ${\ displaystyle A_ {1} (t)}$ and $A_{2}(t)$ ${\ displaystyle A_ {2} (t)}$ ${\ displaystyle A_ {2} (t)}$ change slowly compared to the signal itself ^[2] .

This decomposition underlies the quadrature amplitude modulation (QAM). Based on KAM, such modulations as BPSK and QPSK are created and widely used.

Harmonic

It is known that a linear combination of harmonic oscillations with the same frequency is a harmonic oscillation with the same frequency. The converse is also true: any harmonic signal $S(t)=A\sin(\omega t+\varphi )$ ${\ displaystyle S (t) = A \ sin (\ omega t + \ varphi)}$ $S(t)=A\sin(\omega t+\varphi )$ can be decomposed into the sum of two signals of the same frequency, but shifted in phase. It’s most convenient to take a phase shift by $\pi /2$ ${\ displaystyle \ pi / 2}$ $\pi/2$ . This means that any harmonic oscillation can be represented as the sum of two functions $\cos(\omega t)$ ${\ displaystyle \ cos (\ omega t)}$ $\cos(\omega t)$ and $\cos(\omega t-{\frac {\pi }{2}})=\sin(\omega t)$ ${\ displaystyle \ cos (\ omega t - {\ frac {\ pi} {2}}) = \ sin (\ omega t)}$ $\cos(\omega t-{\frac {\pi }{2}})=\sin(\omega t)$ :

S(t)=A\sin(\omega t+\varphi )=A_{1}\sin(\omega t)+A_{2}\cos(\omega t)

{\ displaystyle S (t) = A \ sin (\ omega t + \ varphi) = A_ {1} \ sin (\ omega t) + A_ {2} \ cos (\ omega t)}

S(t)=A\sin(\omega t+\varphi )=A_{1}\sin(\omega t)+A_{2}\cos(\omega t)

Here $A_{1}=A\cos(\varphi ),A_{2}=A\sin(\varphi )$ ${\ displaystyle A_ {1} = A \ cos (\ varphi), A_ {2} = A \ sin (\ varphi)}$ $A_{1}=A\cos(\varphi ),A_{2}=A\sin(\varphi )$ . This is similar to how a vector in a plane with polar coordinates $(A,\varphi )$ ${\ displaystyle (A, \ varphi)}$ $(A,\varphi )$ decomposes into the sum of two vectors $A_{1}{\vec {x}}+A_{2}{\vec {y}}$ ${\ displaystyle A_ {1} {\ vec {x}} + A_ {2} {\ vec {y}}}$ $A_{1}{\vec {x}}+A_{2}{\vec {y}}$ where $A_{1}=A\cos(\varphi ),A_{2}=A\sin(\varphi )$ ${\ displaystyle A_ {1} = A \ cos (\ varphi), A_ {2} = A \ sin (\ varphi)}$ $A_{1}=A\cos(\varphi ),A_{2}=A\sin(\varphi )$ Are the Cartesian coordinates of the original vector.

Quasi-Harmonic Signal

If the signal is not a pure harmonic signal, but is quasi-harmonic , that is, a signal of the form $S(t)=A(t)sin(\omega t+\varphi (t))$ ${\ displaystyle S (t) = A (t) sin (\ omega t + \ varphi (t))}$ where is the amplitude $A(t)$ ${\ displaystyle A (t)}$ and phase $\varphi (t)$ ${\ displaystyle \ varphi (t)}$ change over time, but not very fast compared to frequency $\omega$ ${\ displaystyle \ omega}$ then we can still decompose $S(t)$ ${\ displaystyle S (t)}$ in the same way:

S(t)=A_{1}(t)\cos(\omega t)+A_{2}(t)\sin(\omega t)

{\ displaystyle S (t) = A_ {1} (t) \ cos (\ omega t) + A_ {2} (t) \ sin (\ omega t)}

But now $A_{1},A_{2}$ ${\ displaystyle A_ {1}, A_ {2}}$ will also depend on time. This is the decomposition into in-phase and quadrature components.

Integrated Envelope

For the concept of the meaning of I / Q decomposition, it is useful to have an idea of the complex envelope . Using Euler's formula , the complex signal $Z(t)=A(t)cos(\omega t+\varphi (t))+jA(t)sin(\omega t+\varphi (t))$ ${\ displaystyle Z (t) = A (t) cos (\ omega t + \ varphi (t)) + jA (t) sin (\ omega t + \ varphi (t))}$ where $j$ ${\ displaystyle j}$ - imaginary unit , can be represented as $Z(t)=A(t)e^{j(\omega t+\varphi (t))}$ ${\ displaystyle Z (t) = A (t) e ^ {j (\ omega t + \ varphi (t))}}$ , and in the case of unequal amplitudes of the sinusoidal and cosine components, we obtain $Z(t)=A_{1}(t)cos(\omega t+\varphi (t))+jA_{2}(t)sin(\omega t+\varphi (t))$ ${\ displaystyle Z (t) = A_ {1} (t) cos (\ omega t + \ varphi (t)) + jA_ {2} (t) sin (\ omega t + \ varphi (t))}$ and then $Z(t)={\sqrt {A_{1}(t)^{2}+A_{2}(t)^{2}}}e^{j(\omega t+\varphi (t))}$ ${\ displaystyle Z (t) = {\ sqrt {A_ {1} (t) ^ {2} + A_ {2} (t) ^ {2}}} e ^ {j (\ omega t + \ varphi (t) )}}$

Quadrature Modulation

The main application of I / Q decomposition is quadrature modulation . Radio signal $S(t)$ ${\ displaystyle S (t)}$ described by such basic parameters as: amplitude $A(t)$ ${\ displaystyle A (t)}$ , the carrier frequency ω and the initial phase φ.

S(t)=A(t)sin(\omega t+\varphi )

{\ displaystyle S (t) = A (t) sin (\ omega t + \ varphi)}

Each of these parameters over time can vary within certain limits. The nature of a change in a parameter may contain information transmitted using a signal. Changing a signal parameter is called modulation . A carrier signal is also distinguished from a modulating signal (one that “overlaps” the carrier). Cosine argument is called full phase $\Phi (t)=\omega t+\varphi$ ${\ displaystyle \ Phi (t) = \ omega t + \ varphi}$ . Thus, we can say that either the amplitude can be modulated $A(t)$ ${\ displaystyle A (t)}$ ( amplitude modulation ) or full phase $\Phi (t)$ ${\ displaystyle \ Phi (t)}$ ( frequency and phase modulation). The carrier frequency of the signal is constant, so when modulating, you can control only two parameters - the amplitude and phase. Based on the foregoing, the signal can be represented as

S(t)=A(t)cos(\omega t+\varphi (t))

{\ displaystyle S (t) = A (t) cos (\ omega t + \ varphi (t))}

The basic idea of quadrature modulation is that the signal $S(t)$ ${\ displaystyle S (t)}$ is represented as the sum of two sinusoidal components whose phase difference is 90 ° (π / 2). The first component: $I=I(t)cos(\omega t)$ ${\ displaystyle I = I (t) cos (\ omega t)}$ . The second component: $Q=Q(t)cos(\omega t+\pi /2)=Q(t)sin(\omega t)$ ${\ displaystyle Q = Q (t) cos (\ omega t + \ pi / 2) = Q (t) sin (\ omega t)}$ . By changing the amplitude of the I / Q components and their further summing, you can get a signal of any kind of modulation.

Notes

↑ Quadrature signal . / Technical Dictionary // Big Technical Encyclopedia
↑ Zyuko A.G., Klovsky D.D., Nazarov M.V., Fink L.M. Theory of signal transmission. - M .: Communication, 1980. - S. 51. - 288 p.

Literature

Gast, Matthew. 802.11 Wireless Networks: The Definitive Guide. - 2. - Sebastopol, CA: O'Reilly Media, 2005-05-02. - Vol. 1. - P. 284. - ISBN 0596100523 .
Franks, LE Signal Theory. - Englewood Cliffs, NJ: Prentice Hall, September 1969. - P. 82. - ISBN 0138100772 .
Steinmetz, Charles Proteus. Lectures on Electrical Engineering. - 1. - Mineola, NY: Dover Publications, 2003-02-20. - Vol. 3.- ISBN 0486495388 .
Steinmetz, Charles Proteus (1917). Theory and Calculations of Electrical Apparatus 6 (1 ed.). New York: McGraw-Hill Book Company. B004G3ZGTM .
Wade, Graham. Signal Coding and Processing. - 2. - Cambridge University Press, 1994-09-30. - Vol. 1. - P. 10. - ISBN 0521412307 .
Naidu, Prabhakar S. Modern Digital Signal Processing: An Introduction. - Pangbourne RG8 8UT, UK: Alpha Science Intl Ltd, November 2003. - P. 29–31. - ISBN 1842651331 .

Links

I / Q Data for Dummies
What is I / Q Data? (eng.)

[БТЭ-1] Quadrature signal . / Technical Dictionary // Big Technical Encyclopedia

[2] Zyuko A.G., Klovsky D.D., Nazarov M.V., Fink L.M. Theory of signal transmission. - M .: Communication, 1980. - S. 51. - 288 p.