Passband Signals

Passband Signals

The term Passband signal in the current context refers to the modulated signal that results from a baseband signal modulating a carrier wave.  Passband signals have some interesting characteristics that we will cover by referring to the diagrams below. (Disclaimer: illustrative purposes only).

Real-valued baseband signal and modulated passband signal plotted on the real frequency axis.

Complex-valued baseband signal and modulated passband signal plotted in the real frequency axis.

Properties of Passband Signals

Shifted Frequency Response

The complete frequency response (including both positive frequencies and negative frequencies) of the baseband signal is preserved in the passband signal, but is now centered around the positive and negative frequencies of the modulated carrier wave.  We say that the baseband frequency response has been “moved up” from 0 Hz to the frequency of the carrier.

Bandwidth “Doubling”.

If we were to describe the bandwidth of the real baseband signal in the first image, we would simply describe it by its positive frequency components (in green).  We don’t stop to include the negative frequency components.  Think about the voice signal in a narrow-band digital phone.  We measure the bandwidth from 0Hz to the maximum significant frequency component, its bandwidth is 3400 Hz.  In comparison, the bandwidth of the passband signal is measured from its smallest significant frequency component to its largest significant frequency component.  This value is double that of the old baseband signal bandwidth.  This is a common phenomenon in all most forms of wireless communications and modulation types involving real signals.

There is an analog amplitude modulation variant called single-sideband modulation that conditions the input and output signals of the modulator using mixers and filters respectively to eliminate the frequency doubling effect.  I never encountered something like this in digital communications technologies.

As shown in the second image, complex baseband signals do not have a symmetrical frequency response around the 0 Hz mark.  As a result, when they are used to modulate a complex carrier,  you will simply see the complete frequency response shifted up to the carrier frequency.  There is no guarantee we will see the same “bandwidth doubling” effect that we do with the symmetrical frequency response of a real valued signal.  That said, the same mechanics are involved that shift the frequency response to the positive and negative carrier frequencies!

Symmetry around 0 Hz.

The real frequency components of a passband signal are symmetrical around the 0 Hz mark.  That means that the real frequency components at negative frequencies are equal to the real frequency components at positive frequencies for the passband signal.  We will not concern ourselves with the imaginary frequency components that occur when dealing with complex signals as our focus is on communications signals which are predominantly real numbered.

If you conduct a Fourier transform on the modulated carrier signal, you will see the real, negative-frequency components of the modulated signal centered around -fc.  You will notice that these are the perfect mirror image of the real, positive-frequency components centered around fc.  If you look at real world version of the same signal through a spectrum analyzer, you will see only the positive-frequency components.

That’s all for now

Rob

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