A visual introduction to concepts such as sampling rates, I/Q data, and modulation techniques that are needed for understanding software-defined radio.

Sampling Rate

I/Q Data

Summation

Any waveform can be constructed from or decomposed into a sum of sine waves. For example, a square wave is an infinite sum of odd harmonics, given by the following equation.

\[ y(t) = \sum_{k=0}^{\infty} \frac{sin(2\pi (2k+1)f t)}{2k+1} \]

Modulation

Every sinusoidal signal has an amplitude \(A\), frequency \(f\), and phase \(\phi\). The value of the sinusoidal signal at time \(t\) is given by the equation:

\[ y(t) = A \cdot cos(2\pi f t + \phi) \]

If omitted from equations, default values of \(A = 1\) and \(\phi = 0\) can be assumed.

Amplitude Modulation (AM)

In amplitude modulation (AM), the amplitude of the modulating signal \(m(t)\) modulates the amplitude of the carrier signal. In the following equation, \(A\) is the carrier amplitude and \(M\) is the modulation amplitude.

\[ y(t) = \Big(A \cdot cos(2\pi f_c t)\Big) \cdot \Big(1 + \Big(\frac{M}{A} \cdot m(t)\Big)\Big) \]

In the following example, the carrier signal \(cos(2\pi f_c t)\) and the modulating signal \(m(t) = cos(2\pi f_m t)\) produce the AM signal:

\[ y(t) = \Big(0.5 \cdot cos(2\pi f_c t)\Big) \cdot \Big(1 + \Big(0.75 \cdot cos(2\pi f_m t)\Big)\Big) \]

Frequency Modulation (FM)

In frequency modulation (FM), the amplitude of the modulating signal \(m(t)\) modulates the frequency of the carrier signal. The actual modulation uses a constant \(k\) that is based on properties of \(m(t)\), and a signal \(\psi(t)\) that is an appropriate invertible transform on \(m(t)\).

\[ y(t) = cos\Big(2\pi f_c t + \Big(k \cdot \psi(t)\Big)\Big) \]

In the following example, the carrier signal \(cos(2\pi f_c t)\) and the modulating signal \(m(t) = cos(2\pi f_m t)\), with \(\psi(t) = sin(2\pi f_m t)\), produce the FM signal:

\[ y(t) = cos\Big( 2\pi f_c t + \Big(\frac{\Delta{f}}{f_m} \cdot sin(2\pi f_m t)\Big) \Big) \]