The Concept of Frequency

A complex sinusoid V rotating in time IQ-plane and generating two real sinusoids

Wireless signals between two devices travel through the use of radio waves. A radio wave is an electromagnetic wave propagated by an antenna. Radio waves have different frequencies, and one can pick up a specific signal by tuning a radio receiver to a specific frequency. But what is a frequency anyway?

A Complex Sinusoid


Consider a complex number V in an IQ-plane with an angle \theta and length A as shown in this Figure. Its projection on I-axis is equal to A \cos\theta and projection on Q-axis is equal to A \sin\theta. Now imagine V rotating with an angle continuously increasing with time. Then, V can be treated as a signal with time as independent variable and we call it a complex sinusoid. Just like velocity is the rate of change of displacement, frequency is the rate of change in phase of a complex sinusoid. This rate of change of phase results in V rotating in the time IQ-plane at an angular velocity 2\pi F, as shown in Figure below.

A complex sinusoid V rotating in time IQ-plane and generating two real sinusoids

Note that as time passes, V is shown in Figure above as coming out of the page. When its projection from a 3-dimensional plane to a 2-dimensional plane formed by time and I-axis is drawn, we get A \cos (2\pi F t + \theta). Similarly, when the projection is drawn on a 2-dimensional plane formed by time and Q-axis, it generates A \sin (2\pi F t + \theta). Randomly choosing \cos(\cdot) as our reference sinusoid, the I component is called inphase because it is in phase with \cos(\cdot) while Q component is called qaudrature because \sin(\cdot) is in quadrature — i.e.,90^ \circ apart — with \cos(\cdot).

In conclusion, a complex sinusoid with frequency F is composed of two real sinusoids

(1)   \begin{align*}             V_I\: &= A\cos(2\pi F t  + \theta) \\             V_Q   &= A\sin(2\pi F t + \theta)         \end{align*}

where F is the continuous frequency with units of cycles/second or Hertz (Hz). The direction of rotation is considered positive for anticlockwise rotation, and negative for clockwise rotation.

Amplitude A and phase \theta are the two other parameters that characterize a sinusoidal signal, where A determines the maximum amplitude for the sinusoid, and \theta determines the initial angular offset of V at t=0.

Sinusoid parameters


Remember that whenever you hear the word “frequency”, it is the frequency of a sinusoid A\cos(2 \pi F t  + \theta) or A\sin(2 \pi F t  + \theta). There are 3 parameters that characterize a sinusoid:

  1. Frequency \rightarrow F
  2. Phase \rightarrow \theta
  3. Amplitude \rightarrow A

Referring to this Figure, it is clear that if the complex sinusoid V rotates faster, the corresponding real sinusoids jump up and down faster, the time T for completing one period becomes smaller, and we call it a higher frequency. Therefore, it is related to time period T as

    \begin{align*}     F = \frac{1}{T}     \end{align*}

and the range of continuous frequency values is

    \begin{equation*}      -\infty < F < \infty     \end{equation*}

Example


When one tunes to a radio station at 88 MHz, one is actually listening to a station broadcasting a radio signal at a carrier frequency of 88\times10^6 Hz, which means that the transmitter is oscillating at a frequency of 88,000,000 cycles/second. Accordingly, that wave is completing one period in T = 1/F = 11.4 ns.

What is a Negative Frequency?


Negative frequencies can cause some confusion as it is hard to visualize a negative frequency viewed as inverse period of a sinusoid. Since the term magnitude is absolute value of the amplitude, the inverse period only gives the magnitude of that frequency. Define it through the rate of rotation of a complex sinusoid V in this Figure, and it is evident that a negative frequency simply implies rotation of V in a clockwise direction. Negative frequencies are real, just like negative numbers are real.

A Real Sinusoid


We have learned that a complex sinusoid rotating in time IQ-plane generates two real sinusoids. The question is: How to produce only one real sinusoid in complex plane?

Interestingly, just like a complex sinusoid is made up of two real sinusoids (Eq (1)), a real sinusoid can be produced by two complex sinusoids rotating in opposite directions to each other, one with a positive frequency F and other with a negative frequency -F. From Figure below, it can be seen that Q components of the complex sinusoids cancel out in this scenario while I parts add up to form a cosine wave. As before, time dimension, not shown here, is coming out of the page.

Two complex sinusoids rotating in opposite directions in time IQ-plane and generating one real sinusoid

From the definitions of a complex sinusoid V in Eq (1) and conjugate of a complex number in this Eq, a real cosine wave can be written as

    \begin{align*}         V_I &= \frac{1}{2}(V + V^*)     \end{align*}

The amplitudes of both complex sinusoids are scaled by 1/2 to cater for factor 2 from addition of two similar I parts. A sine wave can also be constructed in a similar manner.

Frequency Domain


Above, we have talked about frequency being the rate of rotation of a complex sinusoid in time IQ-plane (as in this Figure). It is evident that this rate of rotation can be changed from very slow (close to 0) to as fast as possible (close to +\infty). As also explained above, a clockwise direction of rotation implies a negative frequency, and hence the complete range of frequencies of a complex sinusoid is from -\infty to +\infty. When a signal or function is drawn in frequency domain, the graph actually shows the I and Q components of those complex sinusoids whose frequencies are present in that signal.

Since one complex sinusoid has a single frequency, it is drawn as a narrow impulse on the frequency IQ-plane. Remember that this frequency IQ-plane is different than the time IQ-plane drawn in this Figure earlier.

Let us look at examples of a real cosine wave and a real sine wave in frequency domain representation in Figure below.

A sine and cosine in both time and frequency IQ-planes

Time IQ-plane vs Freq IQ-plane


The arrows in frequency IQ-plane are not rotating like this Figure and this Figure earlier. In fact, these are frequency domain impulse symbols indicating a single spectral line for a single complex sinusoid. The height of the impulses in frequency domain indicates the magnitude of that sinusoid while their relative phases can be seen through the directions in which the spectral impulses are pointing.

Any signal plotted in frequency domain represents the magnitudes and phases of complex sinusoids combined together to form that signal. As such, they form a basic unit of signal construction of any shape. As more and more such sinusoids come together, they form a continuum in frequency domain that is illustrated as a continuous spectral graph.

A natural question arises at this stage: What about the signals having components other than these nice looking sinusoids? The answer is surprising: There are hardly any! Long ago, scientists figured out that most signals of practical interest can be considered as a sum of many sinusoids — possibly infinite — oscillating at different frequencies regardless of the signal shape. So an arbitrary signal s(t) can be written as

    \begin{align*}   s(t) &= a_0 \sin (2\pi F_0 t) + a_1 \sin (2\pi F_1 t) + a_2 \sin (2\pi F_2 t) + \cdots \\        &= a_0 \sin (2\pi\frac{1}{T_0} t) + a_1 \sin (2\pi\frac{1}{T_1} t) + a_2 \sin (2\pi \frac{1}{T_2} t) + \cdots \end{align*}

where a_0, a_1,a_2,\cdots, are amplitudes that determine the impact each sinusoid has on s(t). If s(t) has no sinusoid of frequency F_k, then the corresponding amplitude a_k = 0.

It is difficult to believe in such a statement for many signals with sharp edges like a square or triangular waveform but this concept is true even for such signals. In that case, the number of sinusoids participating in construction of that signal tends to \infty.

As an example, consider a square wave signal. The curves in Figure below show how it is approximated with integer multiples of a fundamental frequency F = 1/T (the corresponding negative axis in frequency domain is not shown for simplicity).

Time and frequency representations of a square wave signal

The blue curve consists of the first two terms

    \begin{align*}       s(t) &= \sin(2 \pi F t) + \frac{1}{3}\sin( 2 \pi 3F t) \\            &= \sin \left(2 \pi\frac{1}{T} t \right) + \frac{1}{3}\sin \left(2 \pi\frac{3}{T} t \right)     \end{align*}

which is clearly quite inaccurate. However, increasing this approximation with just three more terms in red curve as

    \begin{align*}       s(t) &= \sin(2 \pi F t) + \frac{1}{3}\sin(2 \pi 3F t) + \frac{1}{5}\sin(2 \pi 5F t) + \\            & \hspace{2.6in} \frac{1}{7}\sin(2 \pi 7F  t) + \frac{1}{9}\sin( 2 \pi 9F t) \\            &= \sin \left(2 \pi\frac{1}{T} t \right) + \frac{1}{3}\sin \left(2 \pi\frac{3}{T} t \right) + \frac{1}{5}\sin \left(2 \pi\frac{5}{T} t \right) + \\            & \hspace{3in} \frac{1}{7}\sin \left(2 \pi\frac{7}{T} t \right) + \frac{1}{9}\sin \left(2 \pi\frac{9}{T} t \right)     \end{align*}

displays significantly closer behaviour. If we increase the number of sinusoids with respective decreasing amplitudes, we can improve this approximation even more.

Spectrum and Bandwidth


A frequency spectrum, or simply the spectrum, is just the range of all possible frequencies of electromagnetic radiation. A full continuous spectrum, for example, includes radio waves, microwaves, infrared, ultraviolet, x-rays and gamma rays.

In an ideal world, bandwidth of a signal is the range of frequencies of sinusoids present in that signal. In other words, taking into account all the sinusoids, the bandwidth is simply the difference between the highest frequency F_H and the lowest frequency F_L in the spectrum of that signal.

    \begin{equation*}       \textmd{BW} = F_H - F_L     \end{equation*}

An ideal band-limited signal has a spectrum that is zero outside a finite frequency range F_L \le |F| \le F_H:

    \begin{equation*}     S(F) = \left\{ \begin{array}{l}     0, \quad 0 \le |F| \le F_L \\     ?, \quad |F_L| \le |F| \le |F_H| \\     0, \quad F_H \le |F| \le \infty \\     \end{array} \right.     \end{equation*}

Having said that, a signal can have a very low contribution from a sinusoid of a particular frequency but not completely zero. Skipping the mathematical proof, we present the following argument: Remember that the concept of frequency is defined through sinusoids that are infinitely long in time domain, see this Figure. Since sinusoids exist only for a finite duration in real-world, the spectrum of these truncated sinusoids is not an impulse as shown in this Figure. Instead, frequency domain representation of a finite duration sinusoid spreads out in the entire spectrum from -\infty to +\infty.

Time and frequency support


A signal cannot be limited in both time and frequency domains. For practical implementations, a signal must be time limited which makes it unlimited in frequency. Therefore, every real signal occupies an infinite amount of bandwidth. A band-limited signal is then referred to as a signal with most of its energy concentrated within a certain amount of frequency range. Practical definitions of bandwidth vary depending on that amount of energy.

For example, Federal Communications Commission (FCC) defines bandwidth as the band in which 99 \% of the signal power is contained. Another common definition is that everywhere outside the specified band, a certain attenuation (say, 60 dB) must be attained.

As a general rule, signals with fast irregular variations have large bandwidth as a large number of sinusoids are required to build such a signal, while slowly varying signals have low bandwidth. Consequently, signals that are narrow in time domain have a large bandwidth in frequency domain, and vice versa.

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