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Fundamentals of Digital Signal Processing

Phase Rotations

Explore how phase rotations impact complex signals in digital signal processing. Understand the mathematical principles and formulas that describe phase rotation by an angle, and how they alter the in-phase and quadrature parts of signals while preserving magnitude. This lesson clarifies the practical rules for phase rotation essential for DSP applications and hardware implementations.

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Phase rotations impact a signal in a variety of ways. Therefore, understanding how a signal is rotated inphase is important for a DSP learner.

Complex signals

In complex notation, rotating a complex number in the IQIQ plane by a phase θ\theta is very simple.

  • For a number VV and phase θ\theta, we write a phase rotation as:

W=V⋅ejθW= V\cdot e^{j\theta}

Since ejθe^{j\theta} is a complex number with magnitude 1 and phase θ\theta, the magnitude of the result WW stays constant while θ\theta is added to the angle of that complex number.

  • For a signal x(t)x(t) and phase θ\theta, we write a phase rotation as:

y(t)=x(t)⋅ejθy(t)= x(t)\cdot e^{j\theta}

Again, the magnitude of the result y(t)y(t) stays constant. This means that the individual II and QQ values change, but their squared sum remains the same. The impact of θ\theta appears for all time instants.

Real signals

To understand the same process in terms of real signals, let’s multiply a complex number V ...