User:WillWare/Radio theory

Trig identities

There are some basic trigonometric identities that explain the appearance of sum and difference frequencies when a carrier wave is modulated.

sin (x + y) = sin x cos y + sin y cos x

cos (x + y) = cos x cos y - sin x sin y

sin x sin y = (1/2) cos (x - y) - (1/2) cos (x + y)

cos x cos y = (1/2) cos (x + y) - (1/2) cos (x - y)

The mathematical function for a sine wave signal of frequency f is

A sin(2πft + φ)

where A is the magnitude (or loudness) of the signal and φ is the phase (or offset in time).

Amplitude modulation

Put in a time-domain graph of what's going on here, with the unmodulated f₁ carrier, the f₂ signal, and the resulting modulated carrier. It's probably good to show overmodulation. Simple AM receivers are generally insensitive to the phase of the carrier and can't correctly interpret an overmodulated signal.

In amplitude modulation, a carrier wave of frequency f₀ is modulated by an audio frequency f₁ to give a signal like this:

(1/2 + 1/2 sin 2πf₁t) * sin 2πf₀t

= 1/2 sin 2πf₀t + 1/2 sin 2πf₁t sin 2πf₀t

= 1/2 sin 2πf₀t + 1/4 cos 2π(f₀-f₁)t - 1/4 cos 2π(f₀+f₁)t

The first term is a strong sine wave at the original carrier frequency f₀. The other two terms are a lower sideband at frequency f₀-f₁ and an upper sideband at frequency f₀+f₁. Imagine a graph with frequency along the X axis, and a big spike at f₀ and two smaller spikes at f₀-f₁ and f₀+f₁. The instrument that displays this graph is called a spectrum analyzer.

In general the audio signals aren't simple sine waves. They are voices, musical instruments, or other complex sounds, so the audio signal is comprised of large numbers of sine waves. On the spectrum analyzer, this looks like a big spike in the middle at f₀, and fuzzy tails going to the left and right for the sum and difference frequencies. The tails will look symmetric because each frequency component in the audio will contribute equal amounts to the sum and the difference frequencies to the right and left of the carrier.

Bandwidth

The audio signal that modulates the carrier can be written as a sum of sinusoids:

x(t) = Σ_k A_k cos (2πf_kt + φ_k)

This results in lower sidebands with frequencies f₀-f_k and upper sidebands with frequencies f₀+f_k, and if F is the largest value for the audio frequencies f_k then the entire modulated signal lies within a frequency window from f₀-F to f₀+F, with a width of 2F.

We can say that 2F is the bandwidth of the modulated signal. In practice, the exact bandwidth may not be so clearly defined. The maximum modulating frequency can vary although it may be bounded. The edges of the modulated signal may not be sharp, they may slope gradually down to the noise floor of the system. It is fairly common to decide that the bandwidth should be measured at the frequency width where the modulated signal falls a certain number of decibels below the center amplitude, for example, 3 dB, where the transmitted power has fallen by a factor of two.

Single-sideband modulation

Single-sideband modulation attempts to allocate both transmitter power and radio spectrum more efficiently by transmitting no carrier, and only the lower or upper sideband. The only piece of information not contained in the signal is the carrier frequency, and the radio spectrum has been reduced by more than a factor of two.

Sinusoids as complex exponentials

Euler showed that we can define the exponential of an imaginary this way:

e^jx = cos x + j sin x

where j=sqrt(-1) and x is in radians. If we do this, a lot of very consistent and useful mathematics falls out as a result. Everything you ever learned about logs and exponentials still applies, but is now generalizable to complex numbers. In dealing with sine waves as we do in radio, Euler's formula becomes amazingly useful. We'll follow the mathematical conventions that ω=2πf, where f is the frequency of the sine wave.

Any sine wave is characterized by its frequency, amplitude, and phase. Assume the frequency is already known, then we have f and omega handy, and remembering our trig identities, we can write

x(t) = A cos(2πft + B)

= A cos(ωt + B) /* replace f with w */

= (A cos B) cos ωt - (A sin B) sin ωt /* trig identity */

= real_part{ X e^jωt } /* Euler's formula */

where X = A (cos B + j sin B) = A e^jB

If you have a bunch of sinsusoids that are mathematically related and they're all the same frequency, then e^jwt will be the same for all of them. Then it's handy to drop that part and just work with the complex coefficient X.

X is often called a phasor. It has an absolute value equal to the magnitude of the sine wave, and an angle (arctangent(imaginary_part/real_part)) equal to the phase of the sine wave (assuming phase=0 is a cosine).

Impedance

One very handy thing about phasors is that we can take a time derivative by multiplying by jω, or a time integral by dividing by jω. Remember those time-derivative formulas for capacitors and inductors?

 i = C dv/dt  ==>   i = jω C v  ==>  XC = 1 / Cjω

 v = L di/dt  ==>   v = jω L i  ==>  XL = Ljω

If we have an inductor L and we're seeing a sine wave voltage with phasor V and a sine wave current with phasor I, then we have a complex-valued version of Ohm's Law:

 V = I XL

X_C and X_L are impedances. Think of "impedance" as a generalization of "resistance" just the way "complex number" is a generalization of "real number". A pure-real impedance is just a resistance - the voltage and current are in phase with each other. For capacitors and inductors, as for most circuits, impedance varies with frequency.

Summing complex exponentials to get real-valued signals

So Euler told us that

ejωt = cos ωt + j sin ωt

If we have a sinusoidal voltage, it's real-valued (because there are no imaginary electrons) and we can write

cos ωt = 0.5 ejωt + 0.5 e-jωt

This gets rid of a kludge. Now we need to keep track of two different phasors (0.5 for jωt, 0.5 for -jωt) but the operation of taking the real part for no clear reason goes away, and is replaced by an addition of two complex values to get a real value. We can also write

sin ωt = 0.5j ejωt -0.5j e-jωt

So this is another approach to the question of "what are negative frequencies?"

Here, negative frequencies are used to produce complex conjugates, which can be added together to produce real-valued signals on physical wires. The complex numbers are only a mathematical convenience, but things need to be real when we look at real circuits, and negative frequencies are a way to get there with mathematical consistency.