Response to Pete’s Technical Critique
Hi Pete,
Thank you so much for taking the time to provide such detailed technical feedback! I really appreciate your careful reading. You’ve raised some important points that deserve thoughtful responses. Let me try and address each one.
Filtering Out the Lower Image and Complex-Valued Signals
I was imprecise here. My statement “Let’s filter out the lower image, and transmit fc” could be misleading in the context I presented it.
What I was trying to convey to a general audience is the conceptual process that happens in single-sideband modulation when we move to complex baseband representation. What you are saying is that by placing this immediately after showing Euler’s identity for a real cosine, I created confusion about what’s actually transmittable.
You’re correct that if we literally kept only the positive frequency component (α/2)e^(i2πfct), we’d have a complex-valued signal that cannot be transmitted with a single real antenna. In the article’s pedagogical flow, I should have either
1. Been more explicit that this is a conceptual stepping stone to understanding quadrature modulation, or
2. Maintained the real signal throughout this section and introduced the complex representation only when discussing I/Q modulation
Your point is well-taken, and this section could be clarified for technical accuracy.
Sign Convention: I·cos(2πfct) + Q·sin(2πfct) vs. I·cos(2πfct) – Q·sin(2πfct)
This is a really interesting point about convention. You’re right that most DSP textbooks use the negative sign on the sinusoid: I·cos(2πfct) – Q·sin(2πfct), and you’ve identified the key reason. It is consistency with Fourier transform conventions and phase modulation.
For the article’s target audience (QEX readers who are typically radio amateurs with varying levels of DSP background), I made a deliberate choice to use the positive sinusoid form because it’s simpler to explain without being technically wrong.
The positive form I·cos + Q·sin maps more directly to the “cosine on I-axis, sine on Q-axis” geometric intuition that I was building. There is also a hardware convention. The RF engineering texts and datasheets that I have used for I/Q modulators use the positive convention much more often than negative. For readers less comfortable with complex notation, introducing the negative sign requires explaining that it ultimately comes from the complex number properties. This has been a turn-off for people in the past. I did not want it to be a turn off for QEX.
That said, you’re absolutely correct that the negative convention gives cos(2πfct + θ) for phase modulation (standard trig identity). The positive convention gives cos(2πfct – θ), which is non-standard. For frequency modulation, the derivative relationship works correctly only with the negative convention. The negative form aligns with Re{(I + iQ)e^(i2πfct)}
In retrospect I could have used the standard negative convention from the start, or explicitly acknowledged the existence of both conventions and explained why I chose one over the other. Since this is the identical treatment that I got from my grad school notes, I’m not the only person to explain it this way. It is, however, a compromise.
Integration of Cosine from 0 to T Being Zero
This is a significant oversimplification on my part. You’re absolutely right that ∫₀^Tsym cos(2π(2fc)t)dt = sin(2π(2fc)Tsym)/(4πfc), which is generally not zero.
What I meant to convey (but obviously failed to state clearly) is that when we integrate cos(2π(2fc)t) over many symbol periods, or when the symbol period Tsym is chosen to be an integer multiple of the carrier period, the integral approaches zero. In practical systems, the low-pass filter following the mixer rejects the 2fc components.
I should have written something like: “We use a low-pass filter to remove the 2fc terms, leaving us with the DC component from the integration” rather than claiming the integral itself is zero.
This is an important distinction, especially for readers who might try to work through the math themselves. The pedagogical shortcut I took could definitely cause confusion when students try to verify the math or understand DTFT/FFT concepts later. Thank you for the reference to Figure 3-2 in your textbook (which I have). I am sure that many people will take a look.
Image Elimination
You’ve identified another important imprecision. When I wrote “we eliminate an entire image,” I was trying to express that quadrature modulation produces a single-sideband spectrum rather than the double-sideband spectrum of a real carrier. I wanted to stop there because this is something useful in practical radio designs.
Your math clearly shows that the real carrier: I·cos(2πfct) produces symmetric images at ±fc and the quadrature: I·cos(2πfct) + Q·sin(2πfct) = (I/2 + Q/2i)e^(i2πfct) + (I/2 – Q/2i)e^(-i2πfct)
You’re right that this doesn’t automatically or magically eliminate an image. Both positive and negative frequency components exist. However, the key advantage is that with independent control of I and Q, we can create single-sideband modulation where the negative frequency component can be zeroed out.
I should have been more precise: quadrature modulation *enables* single-sideband transmission through appropriate choice of I and Q, rather than automatically eliminating an image.
Feedback like this highlights the challenge of writing about DSP for a mixed audience. Trying to build intuition without sacrificing technical accuracy. I absolutely favor paths towards intuitive explanation I absolutely can and do create technical errors in the process. Being more rigorous with the mathematics, and/or explicitlly noting where there’s a simplication, is something I will take to heart in any future submissions.
I really appreciate you taking the time to work through these details. Our tradition at ORI is to welcome comment and critique. Thank you!
-Michelle Thompson W5NYV