Complex Sinusoids: The Doorway to DSP

Understand how complex sinusoids (the fundamental signals in DSP applications) are constructed in software.

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From our knowledge of complex representation, we know that a complex number V=ejθV=e^{j\theta} in an IQIQ plane is a constant number. Let’s investigate how it changes with time.

A complex signal

Imagine the same complex number V=ejθV=e^{j\theta} rotating counterclockwise in a circle at a constant rate with time as drawn in the figure below. The constant VV now becomes a function of time V(t)V(t) with the time axis in a direction and appears to be coming out of the screen.

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A complex sinusoid as a function of time
A complex sinusoid as a function of time
  • The complex number VV has now become a complex signal V(t)V(t) with time as independent variable. This signal is known as a complex sinusoid.
  • The change in phase Δθ\Delta \theta during an interval Δt\Delta t is a constant known as the angular velocity.

angular velocity=ω=ΔθΔt \begin{equation*} \textmd{angular velocity} =\omega= \frac{\Delta\theta}{\Delta t} \end{equation*}

Let’s explore the real and imaginary parts of this signal.

Inphase and quadrature components

For a complex number V=ejθV=e^{j\theta}, the real and imaginary components are cos⁡θ\cos \theta and sin⁡θ\sin\theta.

  • The projection on x-axis is cos⁡θ\cos \theta.
  • The projection on y-axis is sin⁡θ\sin \theta.

The signal, V(t)V(t) is shown in the figure above and appears to be coming out of the screen.

  • Its projection from a 33-dimensional plane to a 22-dimensional plane downwards gives the real or II part:

VI(t)=cos⁡2πFtV_I(t)= \cos 2\pi F t

  • The projection from a 33-dimensional plane to a 22-dimensional plane on the side gives the imaginary or QQ part:

VQ(t)=sin⁡2πFtV_Q(t)= \sin 2\pi F t

The II part is known as the inphase component and the QQ part is known as the quadrature component. This is due to the convention of choosing cos⁡(⋅)\cos(\cdot) as our reference sinusoid and that sin⁡(⋅)\sin(\cdot) is in quadrature – i.e., 90∘90^ \circ apart with cos⁡(⋅)\cos(\cdot).

Let’s explore this further with the help of some code.

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from mpl_toolkits import mplot3d
import numpy as np
import matplotlib.pyplot as pl
figWidth = 20
figHeight = 10
f = 1.4
A = 0.8
t = np.arange(0, 2.8, 0.01)
iwave = A*np.cos(2*np.pi*f*t)
qwave = A*np.sin(2*np.pi*f*t)
# Plotting
fig = pl.figure(1, figsize=(figWidth,figHeight), constrained_layout=True)
ax = pl.axes(projection='3d')
ax.plot3D(t, iwave, qwave, linewidth=2, color='b')
ax.plot3D(t, iwave, -2*np.ones(len(qwave)), color='r')
ax.plot3D(t, 2*np.ones(len(iwave)), qwave, color='r')
ax.xaxis.set_ticklabels([])
ax.yaxis.set_ticklabels([])
ax.zaxis.set_ticklabels([])
ax.set_xlim(t[0], t[-1])
ax.set_xticks(np.arange(t[0], t[-1], 0.5))
ax.set_ylim(-2, 2)
ax.set_yticks(np.arange(-2, 2, 1))
ax.set_zlim(-2, 2)
ax.set_zticks(np.arange(-2, 2, 1))
ax.tick_params(labelsize=18)
ax.text(0, 0.5, 1, '$V(t)$', fontsize=18)
ax.text(1.5, -3.5, -0.5, '$V_I(t)=\cos (2\pi ft)$', fontsize=18)
ax.text(1.5, 0.6, 2, '$V_Q(t)=\sin(2\pi ft)$', fontsize=18)
# Hide the right and top spines
ax.spines['right'].set_visible(False)
ax.spines['top'].set_visible(False)
ax.set_xlabel("Time", fontsize=18)
ax.set_ylabel("I", fontsize=18)
ax.set_zlabel("Q", fontsize=18)
ax.grid()
pl.savefig('output/complex-sinusoid.png', bbox_inches='tight')

In complex notation, this complex sinusoid is given as:

V(t)=ejωt=ej2πFt\begin{equation*} V(t) = e^{j\omega t}=e^{j2\pi F t} \end{equation*}

Using the fact that quadrature is the perpendicular direction and jj is a rotation by 90∘90^\circ, this can also be written as:

ej2πFt=cos⁡2πFt+jsin⁡2πFt\begin{equation*} e^{j2\pi F t} = \cos {2\pi F t} + j\sin {2\pi F t} \end{equation*}

This is known as Euler’s identity.

Example

When you turn on a microwave on to heat food, it transmits waves at a frequency of 2.4×1092.4\times10^9 Hz, which means that the oscillator frequency is 2,400,000,0002,400,000,000 cycles per second. The period of a wave like this is:

12,400,000,000=0.42 ns\frac{1}{2,400,000,000}=0.42 \text{~ns}

While this electromagnetic wave is real, a complex sinusoid incorporates the phase information in a convenient manner for signal processing. We will see later that complex sinusoids can act as a doorway to understanding signals and DSP operations.