(Preliminary)
(C) Noboru, Ji1NZL Jul.13, 2016
IQ orthogonal demodulator has simple architecture which is useful to use in SDR (Software Defined Radio) receivers. This document describes about it simply to understand it.
1. Architecture
Fig. 1 is the architecture of “IQ orthogonal demodulator” which has a “2 phase orthogonal oscillator". “IQ orthogonal demodulator” here inputs various kinds of modulated RF signal and output (I, Q) signals on the complex number plane.
1. Architecture
Fig. 1 is the architecture of “IQ orthogonal demodulator” which has a “2 phase orthogonal oscillator". “IQ orthogonal demodulator” here inputs various kinds of modulated RF signal and output (I, Q) signals on the complex number plane.
2. How it works
To simplify the explanation, I assume it receives RF sine wave and what signals go out to output (I, Q).
Vin = Vr*sin(ωr*t) …(1)
Vosc = Vc*sin(ωc*t) …(2)
V2 = Vc*sin(ωc*t - π/2) = -Vc*cos(ωc*t) …(3)
Get V3 by multiplying Vin and Vosc.
V3 = Vin*Vosc = Vr*sin(ωr*t) * Vc*sin(ωc*t)
= Vr*Vc*{ sin(ωr*t) *sin(ωc*t) }
= (-1/2)*Vr*Vc*{ cos((ωr+ωc)*t) - cos((ωr-ωc)*t)) } …(4)
Assume using of Down-converted signal for the equation (4).
Then LPF can remove cos((ωr+ωc)*t from (4)
by specifying cut off frequency of LPF to be set to ωcut < ωr+ωc .
Then
V-i = (1/2)*Vr*Vc*cos((ωr-ωc)*t)) …(5)
Get V4 by multiplying Vin and V2.
V3 = Vin*V2 = Vr*sin(ωr*t) * (-Vc*cos(ωc*t))
= -Vr*Vc*{ sin(ωr*t) *cos(ωc*t) }
= (-1/2)*Vr*Vc*{ sin((ωr+ωc)*t) + sin((ωr-ωc)*t)) } …(6)
in (6)., sin((ωr+ωc)*t) can be removed by LPF, then
V-q = (-1/2)*Vr*Vc*sin((ωr-ωc)*t)) …(7)
Thus output voltage signal comes out as V-i and V-q in (5) and (7).
Viewing (5) and (7), they are orthogonal function for time variable “t”.
Shift -π/2 [rad] of V-i by the equation (7)
∵ Shift -π/2 of Vq = (-1/2)*Vr*Vc*sin((ωr-ωc)*t)-π/2)
= (1/2)*Vr*Vc*cos((ωr-ωc)*t)) = V-i …(8)
The equation (8) means (I, Q) = (V-i, V-q) orthogonal signals each other on voltages of the complex number plane.
These orthogonal signals (I, Q) are extracted from the “IQ orthogonal demodulator” here.
3. Examples of how the modulated signals is recovered to “baseband signal”
(a) Receiving SSB (USB)
Assume USB signal Vin = Vrf*sin(ωc+ωs)*t) comes in.
Input sine wave ωc+ωs instead of ωr, ωr := ωc+ωs
Then the equation (5) becomes
V-i = (1/2)*Vr*Vc*cos((ωc+ωs)-ωc)*t)) = (1/2)*Vr*Vc*cos(ωs*t) …(9)
and the equation (7) becomes
V-q = (-1/2)*Vr*Vc*sin(((ωc+ωs)-ωc)*t)) = (-1/2)*Vr*Vc*sin(ωs*t) …(10)
The equation (10) means “baseband signal” Vr*sin(ωs*t) is
demodulated/recovered
as the output voltage V-i and that V-q (10) is orthogonal signal for V-i.
Vq (10) recovered the baseband signal then Voltage Vq can be input to the
audio amplifier directly to sound the speaker.
This means USB is demodulated on V-q output without signal processing by MPU.
V-i (9) is orthogonal signal that shifted -π/2 [rad] of V-q (10).
Here -90 deg. shifted is connected after V-i then (9) becomes
(1/2)*Vr*Vc*cos(ωs*t -π/2) = (1/2)*Vr*Vc*sin(ωs*t) …(11)
This (11) is the same as the inverted signal of (10) so we can add a adder
after (11).
Then (10) + (11) = {(-1/2)*Vr*Vc*sin(ωs*t)} + {(-1/2)*Vr*Vc*sin(ωs*t)}
= -Vr*Vc*sin(ωs*t) … (12)
This signal is two times of Vq at the equation (10) and can be input to
the audio amplifier directly to sound the speaker.
Signals (9), (10) or (12) may be digitized by A/D convertor and they can
be filtered by the filter program to eliminate the extra signal around
demodulated baseband signal.
(b) Receiving SSB (LSB)
Assume LSB signal Vin = Vrf*sin(ωc-ωs)*t) comes in.
Input ωc-ωs instead of ωr, ωr := ωc-ωs
Then the equation (5) becomes
V-i = (1/2)*Vr*Vc*cos((ωc-ωs)-ωc)*t)) = (1/2)*Vr*Vc*cos(-ωs*t)
= (1/2)*Vr*Vc*cos(ωs*t)…(13)
and the equation (7) becomes
V-q = (-1/2)*Vr*Vc*sin(((ωc-ωs)-ωc)*t)) = (-1/2)*Vr*Vc*sin(-ωs*t)
=(1/2)*Vr*Vc*sin(ωs*t)…(14)
The equation (14) means baseband signal Vr*sin(ωs*t) is demodulated in the
output voltage V-q and (14) is orthogonal signal of for V-i.
(14) is recovered the baseband signal. Then Voltage Vq can be input to the
audio amplifier directly to sound the speaker.
This means LSB is demodulated on V-q output.
(14) is orthogonal signal that shifted -π/2 [rad] of V-i.
Here -90 deg. shifted is connected after V-i then (13) becomes
(1/2)*Vr*Vc*cos(ωs*t -π/2) = (1/2)*Vr*Vc*sin(ωs*t) …(15)
This is the same signal as (14) so we can add a adder after (14) and (15)
Then (14) + (15) = {(1/2)*Vr*Vc*sin(ωs*t)} + {(1/2)*Vr*Vc*sin(ωs*t)}
= Vr*Vc*sin(ωs*t) … (16)
This signal is two times of Vq at the equation (14) and can be input to the audio amplifier directly to sound the speaker.
Signals (14) or (15) or (16) may be digitized by A/D convertor and they may be filtered by the filter program to eliminate the extra signal around
baseband.
(c) Receiving AM
AM RF signal is equivalent to the added signals of Carrier ωc, USB ωc+ωs.
i.e. 100% modulated AM signal is expressed as sum of these 3 voltages.
Vin = Vrf*sin(ωc+ωs)*t) + 2*Vrf*sin(ωc)*t) + Vrf*sin(ωc-ωs)*t)
The Carrier signals cancelled as 0 Hz in the demodulator by the equation (5)
and (7).
The Carrier signals is changed to 0Hz. It means there is no tone for the
Carrier signal.
Assume ωr := ωc then (5) becomes
V-i = (1/2)*Vr*Vc*cos((ωc-ωc)*t)) ) = (1/2)*Vr*Vc …(17)
Beside from (7)
V-q = (-1/2)*Vr*Vc*sin((ωc-ωc)*t)) = 0 …(18)
At the output of V-i side,
from (5), re-write it here, V-i = (1/2)*Vr*Vc*cos((ωr-ωc)*t)) {…(5)}
for USB : ωr := ωc+ωs then
V-i for USB becomes Vi= (1/2)*Vr*Vc*cos((ωr-ωc)*t))
= (1/2)*Vr*Vc*cos(( (ωc+ωs) -ωc)*t))
=(1/2)*Vr*Vc*cos(ωs*t) …(19)
for LSB : ωr := ωc-ωs then
Vi= (1/2)*Vr*Vc*cos((ωr-ωc)*t))
= (1/2)*Vr*Vc*cos(( (ωc-ωs)-ωc)*t))
=(1/2)*Vr*Vc*cos(-ωs*t)) = (1/2)*Vr*Vc*cos(ωs*t) …(20)
At the output V-i side, both demodulated LSB and USB are appears, then
V-i = (19) + (20)
= (1/2)*Vr*Vc*cos(ωs*t) + (1/2)*Vr*Vc*cos(ωs*t) = Vr*Vc*cos(ωs*t) …(21)
At the output of V-q side,
from (7) re-write (7) here, V-q = (-1/2)*Vr*Vc*sin((ωr-ωc)*t)) {…(7)}
For USB : ωr := ωc+ωs then
V-q = (-1/2)*Vr*Vc*sin((ωr-ωc)*t))
= (-1/2)*Vr*Vc*sin(( (ωc+ωs)-ωc)*t))
= (-1/2)*Vr*Vc*sin(ωs*t)) …(22)
For LSB : ωr := ωc-ωs then
V-q = (-1/2)*Vr*Vc*sin((ωr-ωc)*t))
= (-1/2)*Vr*Vc*sin(( (ωc-ωs)-ωc)*t))
= (-1/2)*Vr*Vc*sin(-ωs*t))
= (1/2)*Vr*Vc*sin(ωs*t) …(23)
At the output V-q side, both demodulated USB and LSB signals appears, then
V-q = (22)+(23)
= (-1/2)*Vr*Vc*sin(ωs*t) + (1/2)*Vr*Vc*sin(ωs*t) = 0 …(24)
… USB and LSB are cancelled in V-q side and V-q become 0 V.
V-i = (1/2)*Vr*Vc + Vr*Vc*cos(ωs*t) …(25)
Vq = 0 …(26)
Remove DC voltage value from (25) by some condenser such as 1uF or subtracting calculation by MPU.
Then AM signal is decoded as the signal Vr*Vc*cos(ωs*t) …(25)’
Mathematically, Vq = 0 from (24). Demodulated USB and LSB may not be cancelled completely. So we may not use the signal V-q when we receive AM signal.
AM signal may change the phase in some distant signal path and the equation (25) may have some phase angle on the complex plane. This means we have signal strength change according to the phase change. It can be cancelled by the phase angle rotation to become 0. It is the similar principle of so called “Synchronized AM detector”.
(d) Receiving CW (Continuous wave)
CW is equivalent signal of a singular case of LSB where ωs=800Hz single tone or a singular case of USB where ωs=800Hz.
Then CW RF wave can be demodulated as ωs=800Hz single tone on USB side or LSB side.
(e) BPSK, QPSK, and QAM for digital communication
{BPSK: Binary Phase Shift Keying, QAM: Quadrature Amplitude and phase Modulation}
To simplify the explanation, I assume receiving RF sine wave with phase shifted φ(t) and what signals go out to output (I, Q).
Vin = Vr*sin(ωc*t+φ(t)) …(e1)
here φ(t) = 45 deg., 135 deg., 225deg., or 315deg.
= π/4, π/3, π/5, or π/7 are used if 4 values of QAM (=QPSK) are assumed.
Vosc = Vc*sin(ωc*t) …(e2)
V2 = Vc*sin(ωc*t - π/2) = -Vc*cos(ωc*t) …(e3)
Get V3 by multiplying Vin and Vosc.
V3 = Vin*Vosc = Vr*sin(ωc*t+φ(t)) * Vc*sin(ωc*t)
= Vr*Vc*{ sin(ωc*t+φ(t)) *sin(ωc*t) }
= (-1/2)*Vr*Vc*{ cos((2*ωc)*t+φ(t)) - cosφ(t) }
= ( 1/2)*Vr*Vc*{ -cos((2*ωc)*t+φ(t)) + cosφ(t) } …(e4)
Assume using of Down-converted signal for the equation (e4).
Then LPF can remove cos((2*ωc)*t +φ(t)) from (e4) if cut off frequency of LPF is set to ωcut < 2*ωc .
Then
V-i = (1/2)*Vr*Vc*cosφ(t) …(e5)
Get V4 by multiplying Vin and V2.
V3 = Vin*V2 = Vr*sin(ωc*t+φ(t)) * (-Vc*cos(ωc*t))
= -Vr*Vc*{ sin(ωc*t+φ(t)) *cos(ωc*t) }
= (-1/2)*Vr*Vc*{ sin(2*ωc*t+φ(t)) + sinφ(t) } …(e6)
in (6)., sin(2*ωc*t+φ(t)) can be removed by LPF, then
V-q = (-1/2)*Vr*Vc*sinφ …(e7)
Thus output voltage signal comes out as V-i and V-q in (e5) and (e7).
Viewing (e5) and (e7), they are orthogonal function for time t.
Shift -π/2 [rad] of Vi by the equation (e7).
∵ Shift -π/2 of Vq = (-1/2)*Vr*Vc*sin(φ(t)-π/2)
= (1/2)*Vr*Vc*cosφ(t) = V-i …(e8)
The equation (e8) means (I, Q) = (V-i, V-q) orthogonal signals on voltages of the complex number plane, and that they are extracted from the “IQ orthogonal demodulator”.
Now we get φ(t) from the output signal V-i (e5) and V-q (e7) by calculations.
V-q/V-i = (-1/2)*Vr*Vc*sinφ / (1/2)*Vr*Vc*cosφ(t) = -tanφ(t) …(e9)
∴ φ(t) = arctan( - (V-q/V-i) ) …(e10)
Phase value φ(t) of baseband is extracted by the calculation here.
Besides, we get a half length of radius r(t) from V-i (e5) and V-q (e7).
r(t) = √(V-i^2+V-q^2) = √{ {(1/2)*Vr*Vc*cosφ(t) }^2 + {(-1/2)*Vr*Vc*sinφ }^2 }
= √{ {(1/2)*Vr*Vc}^2*cosφ(t)^2 } + {(-1/2)*Vr*Vc}^2*sinφ^2 }
= √{ {(1/2)*Vr*Vc}^2* (cosφ(t))^2 + (sinφ)^2 } }
= (1/2)*Vr*Vc …(e11)
a half of radius value of baseband is extracted by the calculation here.
Demodulating QAM signals require to calculate (e9), (e10), and (e11).
They can be calculated by MPU program for digitalized signals of V-i and V-q.
The principle itself doesn’t give limitation to usages only to QPSK (4 digits values) and it can be expanded to 2^n values transmutation/reception.
( The integer “n” is greater than equals 1 and it has no limitation as Math.
i.e. Values = 2, 4, 8, 16, 32, 64, 128, 256, 512, 1024, 2018, … )
In the actual work, The phase φ(t) of RF QAM at transmitter side and the phase of OSC Vosc at IQ demodulator side is not synchronized. Because QAM itself has no synchronization protocol and transmitters and receiver are located in the distant place.
There is no synchronization mechanism for phase of QAM itself.
So synchronizing signals for the initial phase of QAM and the procedures in the MPU calculation for phase synchronization is required.
(f) Demodulating FM (To be continued)
4. Summary
(1) SSB(USB/LSB), AM, and CW can be demodulated without calculation by MPU.
MPU can do DSP Filtering extra signal around these signal.
(2) QAM can be demodulated with calculations by MPU or DSP via IQ orthogonal demodulator.
(3) IQ orthogonal demodulator can demodulate input of RF signal (Vr-i(t), Vr-q(t)) to convert to output signals (V-i(t), V-q(t)) that are orthogonal functions for the variable time “t” on the complex number plane for voltage.
(4) The principle and architecture is very simple but IQ orthogonal demodulator can be used for many kinds of signal modulation. It has good characteristics to connect MPU or DSP processing. There may be some unknown applications that has not been found yet.
5. Subject
(1) Low tone of beat in AM and strange sound of music on SSB
This method requires very accurate and stable frequency for the local oscillator. If the frequency error exists, the error of the frequency may cause low tone of beat when receiving AM signal. Human’s ear is very sensitive for around 8Hz frequency drifted.
It is not so serious in listening human’s voice but sound of music may be strange or uncomfortable feeling even if turned frequency has very small errors such as 10Hz.
If some PLL as a local oscillator work in every 100Hz steps, it is out of usage to listen to the music in SSB/AM. SSB/AM can not be suitable to send or receive the tone of music.
Old fashioned LC analog oscillator often drifts the frequency gradually by time passed.
However, PLL, DDS , TCXO or latest devices are changing the situation be better rapidly.
(2) Strength of decoded AM signals changes by amount of shifted phase of Oscillator
V-q becomes 0 in the equation (24). This means output of “product detector” can be 0V and radio listener may loose the existence of the radio station’s signal when the Oscillator’s phase shift becomes ±90 deg, ±270deg. (=π/2 [rad], 3π/2).
In general, Carriers of transmitter’s of radio stations and radio’s oscillators can’t be synchronized and signal strength Vr changes from -1*Vr to +1*Vr. This subject have not always known very well.
Almost of radios in the market may not take care of this phase-shift subject in their designs. So called “AM synchronized detector” can solve it. But many radios may have had this potential subject even now.
Traditional diode detector radios don’t have this subject because it works by the carrier that is derived from received original carrier (not shifted) but this method has the subject that 2nd, 3rd, … N-nd harmonics distortions for the baseband signal are generated in it.
(3) Canceling the fading phenomenon by phase change by time
SSB (USB/LSB) signals don’t have carrier on them. So there is no way to canceling fading. But if we add weak carrier on them we can use it to fix this problem.
As AM signal have a carrier, it appears as phase angle on the complex plane.
We can cancel it by DSP calculations. It is similar as so called “AM synchronized detector” .
This “phase-shift” subject wrote on (2) must occur in transfer path between radio stations and this phenomenon have been so called “fading”, but it has not been fixed yet. It also happens in the ionized layer (D, E, and F layers) paths.
If radios have a function to shift phase of oscillator’s frequency, performance of radios can be improved.
(4) Synchronizing phase for QAM
Phase of QAM signal is changed by the change of signal path and by the difference of initial phase of transmitter side and receiver side.
Synchronization for them is required.
(5) Avoiding multi-path distortion for QAM
OFDM is designed to avoid multi-path distortion by using multi-sub-channels of carriers that are orthogonal function for time variable “t”. It may be difficult to remove multi-path distortion of QAM that has single channel of carrier by DSP operations at the current technology. Because it takes long time to process to remove duplicated multi-path distorted signals.
It may be difficult to send fast digital signal on narrow band of HF. So degreasing speed of BPS may be some kind of solution for this subject. Or we may define some small multi-sub-channels of carriers that are orthogonal on the complex plane signal.
Appendix;
A. Actual work of SSB demodulation
B. An example of assembling of "IQ demodulator"
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Revision:
Preliminary public version : Jul.13, 2016 ;
This version may still contain some editorial, mathematical, or technical problems.
I recommend you read my stories carefully. This is under construction :)
(Return to index)
Revision:
Preliminary public version : Jul.13, 2016 ;
This version may still contain some editorial, mathematical, or technical problems.
I recommend you read my stories carefully. This is under construction :)
(Return to index)
