Differences and Similarities of Low-Pass and Band-Pass Signals

# OVERVIEW OF THE FOURTH MAIN CHAPTER #

In the third main chapter ⇒ Aperiodic Signals – Impulses often low-pass signals were assumed, i.e., those signals whose spectral functions lie in the range around the frequency $f = 0$. Particularly in optical transmission and in radio transmission systems - but not only in these - the transmitted signals are in the range around a carrier frequency $f_{\rm T}$. Such signals are called $\text{Bandpass Signals}$.

All principles of the Fourier transform and the inverse Fourier transform described in the last chapter apply to band-pass signals in the same way. Besides there are some special features of band-pass signals, whose observance can lead to a simpler description.

This chapter contains in detail:

the enumeration of differences and "similarities" of low-pass and band-pass signals,
the "synthesis" of band-pass signals from the equivalent low-pass signal,
the&nb "Equivalent low-pass signal" in the time and frequency domain, and finally
the representation of analytical signal/equivalent low-pass signal in the complex plane.

Motivation of Band-Pass Signals for Communications Engineering

In the previous chapters of this book, only signals whose spectra lie in a narrow range around the frequency $f = 0$ have been considered. Examples are analog speech, music and image signals, which all - despite their different bandwidths - can be described as 'low-pass signals .

If you want to transmit such a low-pass signal to a spatially distant sink, the signal may have to be converted to another frequency position. There can be several reasons for this:

Often the transmission channel is unsuitable for the direct transmission of the source signal in the original frequency band, because this band contains frequencies that are unfavorable for it. Only by a frequency shift by means of a so-called modulator a transmission is made possible.
A single transmission channel can be used for the simultaneous transmission of several signals, if they are modulated with different carrier frequencies at the transmitting end. This method is called FDMA ,abbreviated as Frequency Division Multiple Access
The transmission quality can be improved compared to the simplest analog methodAmplitude Modulation at the expense of a larger bandwidth and thus a higher signal-to-noise ratio can be achieved. Examples are the. Frequency Modulation (FM) as an analog method and the digital Pul (PCM).

$\text{Remember:}$ The transmitted signals of many transmission methods are band-pass signals.

Note: Dem Autor ist durchaus bewusst, dass es nach der letzten Rechtschreibreform „Tiefpasssignal” und „Bandpasssignal” heißen müsste. Um diese unschönen Konstrukte zu vermeiden, verwenden wir im Folgenden meist die Schreibweisen „Tiefpass–Signal” und „Bandpass–Signal”, manchmal auch „TP–Signal” und „BP–Signal”.

$\text{Example 1 : To classify signals with respect to „low-pass” and „band-pass”}$

(a) speech and music are low-pass signals with a bandwidth of $\text{20 kHz}$ (at very good quality). Since a radio transmission is only possible from approx. $\text{100 kHz}$ , a conversion to carrier frequencies between

$\text{0.525 ... 1.61 MHz}$ $($Medium wave broadcasting, amplitude modulation, channel spacing $\text{9 kHz})$,
$\text{87.5 ... 108 MHz}$ $($Radio on FM, frequency modulation, channel spacing $\text{300 kHz})$

is needed.

(b) TV image signals have a larger bandwidth, for example $\text{5 MHz}$ . Here, as well, a frequency band shift occurs before the sound and image transmission due to carrier frequencies between

$\text{41 ... 68 / 174 ... 230 MHz}$ (television, VHF band, channel spacing $\text{7 MHz})$,
$\text{470 ... 850 MHz}$ $($television, UHF band, channel spacing $\text{8 MHz})$.

(c) With GSM mobile radio the carrier frequencies in the D-band are $\text{900 MHz}$ and in the D-band $\text{1800 MHz}$.

(d) With optical transmission the electrical signals are converted into light, i.e. to frequencies between ca. $\text{200 THz}$ and $\text{350 THz}$ $($correspondingly $\text{1.55 µm ... 0.85 µm}$ Wavelength).

Properties of BP-Signals

On this page - without claiming to be complete - some characteristics of band-pass signals are compiled and compared to low-pass signals. We start from the spectral functions $X_{\rm TP}(f)$ and $X_{\rm BP}(f)$ according to the following sketch.

Low-Pass and Band-Pass Spectrum

Regarding the graphic is to be remarked:

The triangular shape of the displayed spectra is to be understood schematically and is only to mark the occupied frequency band. So it should not be concluded that all frequencies within the band are actually occupied and that all spectral functions increase linearly with frequency $f$ .
The corresponding time functions $x_{\rm TP}(t)$ and $x_{\rm BP}(t)$ are real for the time being. This means that according to the Mapping Theorem the spectral functions $X_{\rm TP}(f)$ and $X_{\rm BP}(f)$ - related to the frequency $f = 0$ - each have an even real part and an odd imaginary part.
As bandwidth $B_{\rm TP}$ or. $B_{\rm BP}$ for both low-pass and band-pass we refer to the occupied frequency band at the positive frequencies (in the graphic: continuous curves).
As bandwidth $B_{\rm TP}$ or. $B_{\rm BP}$ for low-pass and band-pass we equally denote the occupied frequency band at the positive frequencies (in the graphic: continuous curves).

$\text{Example 2:}$ An example with discrete spectral lines follows. The left graph shows the spectrum $Q(f)$ of the message signal

$$q(t) = 3\hspace{0.05cm}{\rm V} + 4\hspace{0.05cm}{\rm V} \cdot \cos (2 \pi \cdot 3\hspace{0.05cm}{\rm kHz} \cdot t) + 2\hspace{0.05cm}{\rm V} \cdot \sin (2 \pi \cdot 4\hspace{0.05cm}{\rm kHz} \cdot t). $$

The discrete spectral lines of the real part ⇒ ${\rm Re}\big[Q(f)\big]$ are shown in blue and those of the imaginary part ⇒ ${\rm Im}\big[Q(f)\big]$ in red.

Example of a Low-Pass and Band-Pass Spectrum

On the right is the spectrum $S(f)$ after single sideband amplitude modulation (ESB-AM) with the carrier frequency $f_{\rm T} = 100 \,\text{kHz}$. A description of this transmission system can be found in chapter Envelope Demodulation of the book „Modulation Methods”.

According to this description, $q(t)$ is uniquely a low-pass signal and $s(t)$ is a band-pass signal. The bandwidths are $B_{\rm TP} = B_{\rm BP} = 4 \,\text{kHz}$.
If the DC component $(3 \,\text{V})$ would be missing in the source signal, then $q(t)$ would still be described as low-pass filter-like.
Without knowledge of the task, one could interpret $q(t)$ but then also as a band-pass signal with the bandwidth $B_{\rm BP} = 1 \,\text{kHz}$ .

This example shows that there is no clear mathematical distinction between low-pass and band-pass signals.

Description of a BP-Signal with TP-Signals

We consider two low-pass spectra $X_1(f)$ and $X_2(f)$ with the bandwidths $B_1$ and $B_2$ corresponding to the left graph.

Erzeugung eines Bandpass-Spektrums aus Tiefpass-Spektren

You can see from this diagram:

If $X_1(f)$ and $X_2(f)$ up to a frequency $f_{12}$ are identical, the difference describes a band-pass spectrum with bandwidth $B_{\rm BP} = B_1 - f_{12}$. According to the graphic on the right, the following then applies

$$X_{\rm BP}(f) = X_1(f) -X_2(f).$$

Due to the linearity of the Fourier transform, the time function associated with the band-pass spectrum $X_{\rm BP}(f)$ is valid

$$x_{\rm BP}(t) = x_1(t) - x_2(t).$$

It generally follows from the Fourier transformation that the integral over the time function is equal to the spectral value at $f = 0$ . Consequently, this integral is always zero for every band-pass signal:

$$\int_{- \infty}^{+\infty}x_{\rm BP}(t)\hspace{0.1cm}{\rm d}t = X_{\rm BP}(f \hspace{-0.1cm}= \hspace{-0.1cm} 0) =0.$$

$\text{Example 3:}$ The red curves in the two graphs show the band-pass spectrum $X_{\rm BP}(f)$ and the corresponding time function

$$x_{\rm BP}(t) = 10\hspace{0.05cm}{\rm V} \cdot {\rm si} ( \pi \cdot 10 \hspace{0.05cm}{\rm kHz} \cdot t) \cdot {\rm si} ( \pi \cdot 2 \hspace{0.05cm}{\rm kHz} \cdot t) - 2\hspace{0.05cm}{\rm V} \cdot {\rm si} ( \pi \cdot 2 \hspace{0.05cm}{\rm kHz} \cdot t).$$

Low-Pass and Band-Pass Spectrum and Their Signals

Also shown are the two low-pass spectra and signals. You can see from these pictures:

The blue-dotted curve in the left graph represents the trapezoidal spectrum $X_1(f)$ where the equivalent bandwidth $\Delta f_1= 10 \,\text{kHz}$ and the rolloff factor $r_1 = 0.2$ is
The blue-dotted curve in the right graphic shows the corresponding low-pass–signal $x_1(t)$. The signal value at $t = 0$ corresponds to the blue trapezoidal area of the spectrum $X_1(f)$:

$$x_1(t = 0) = 10 \,\text{V}.$$

The green curve applies to the rectangular spectrum $X_2(f)$ with the equivalent bandwidth $\Delta f_2= 2 \,\text{kHz}$. The corresponding time signal $x_2(t)$ runs $\sin(x)/x$-shaped and is valid:

$$x_2(t = 0) = 2 \,\text{V}.$$

The red curve for the band-pass-like signal is the difference between the blue and green curve on the left and right. Accordingly

$$x_{\rm BP}(t = 0) = x_1(t = 0) - x_2(t = 0) = 8 \,\text{V},$$

$$\int_{- \infty}^{+\infty}x_{\rm BP}(t)\hspace{0.1cm}{\rm d}t = X_{\rm BP}(f \hspace{-0.1cm}= \hspace{-0.1cm} 0) =0.$$

BP-Signal Synthesis from Equivalent TP-Signals

We consider a low-pass signal $x_{\rm TP}(t)$ with spectrum $X_{\rm TP}(f)$ according to the left sketch.

If this signal is multiplied by a (dimensionless) harmonic oscillation

$$z(t) = {\cos} ( 2\pi \cdot f_{\rm T} \cdot t)\hspace{0.15cm}\circ\!\!-\!\!\!-\!\!\!-\!\!\bullet\,\hspace{0.15cm} Z(f) = {1}/{2}\cdot \delta (f - f_{\rm T})+ {1}/{2}\cdot \delta (f + f_{\rm T}),$$

the convolution theorem for the spectrum of the signal $x_{\rm BP}(t) = x_{\rm TP}(t) yields \cdot z(t)$:

$$X_{\rm BP}(f) = X_{\rm TP}(f)\star Z(f) = {1}/{2}\cdot X_{\rm TP} (f - f_{\rm T})+ {1}/{2}\cdot X_{\rm TP}(f + f_{\rm T}).$$

Here it is considered that the Convolution of the spectral function $X_{\rm TP}(f)$ with the frequency-shifted Dirac signal $\delta (f - f_\rm {T})$ yields the same spectral function $X_{\rm TP}(f-f_\rm {T})$ shifted to the right by $f_\rm {T}$ .

Ein BP–Spektrum ergibt sich durch beidseitiges Verschieben eines TP–Spektrums

From the right spectral range display you can clearly see that

$$x_{\rm BP}(t) = x_{\rm TP}(t) \cdot {\cos} ( 2\pi \cdot f_{\rm T} \cdot t)$$

is a band-pass signal. The envelope of $x_{\rm BP}(t)$ given by the magnitude of $|x_{\rm TP}(t)|$ . For example, this principle is applied to the amplitude modulation without carrier,which is discussed thoroughly in the book „Modulation Methods”.

You can see from the above graphic:

The spectrum $X_{\rm BP}(f)$ has the same form as $f_{\rm T}$ in the range around the carrier frequency $f_{\rm TP}(f)$ in the range around $f = 0$, but is attenuated with respect to the low-pass spectrum by the factor $2$ .
Since $X_{\rm TP}(f)$ relative to $f = 0$ has an even real and an odd imaginary part, the band-pass spectrum $X_{\rm BP}(f)$ has the same symmetry properties - but now relative to the carrier frequency $f_{\rm T}$.
Also $X_{\rm BP}(f)$ has portions at negative frequencies. Since the corresponding signal $x_{\rm BP}(t)$ is also real according to the above equation, $X_{\rm BP}(f)$ must also have an even real and an odd imaginary part with respect to the frequency $f = 0$ .
The bandwidth of the band-pass signal is twice that of the low-pass signal: $B_{\rm BP} = 2 \cdot B_{\rm TP}$. Prerequisite for the validity of this statement is that the carrier frequency $f_{\rm T}$ is at least larger than the maximum frequency $2$ by the factor $(B_{\rm TP})$ of the signal $x_{\rm TP}(t)$.

$\text{Example 4:}$ A low-pass signal has discrete spectral components at $f_1 = 1\,\text{ kHz}, \,f_2 = 2\,\text{ kHz}, \,f_3 = 3\,\text{ kHz}$ and $f_4 = 4\,\text{ kHz}$:

$$x_{\rm TP}(t) = 0.26\cdot {\cos} ( \omega_1 \hspace{0.05cm} t + 20^{ \circ}) \hspace{0.18cm}+ 0.54\cdot {\cos} ( \omega_2 \hspace{0.05cm} t - 180^{ \circ}) + 0.30\cdot {\cos} ( \omega_3 \hspace{0.05cm} t + 120^{ \circ}) +0.14\cdot {\cos} ( \omega_4 \hspace{0.05cm} t -40^{ \circ}).$$

The corresponding spectrum $X_{\rm TP}(f)$ is complex because of the non-zero phase positions.

DSB-AM-Signal With Different Carrier Frequencies

If one multiplies $x_{\rm TP}(t)$ with a cosine signal of amplitude $1$ and frequency $f_{\rm T} = 20 \,\text{kHz}$, the band-pass signal is obtained according to the upper graphic.
The lower sketch applies to the band-pass signal with the carrier frequency $f_{\rm T} = 100 \,\text{kHz}$.
In both illustrations the function curves $\pm \vert x_{\rm TP}(t) \vert $ can be recognized as the envelope of the band-pass signals.

Notes: The topic of this chapter is covered in the german learning video properties of band-pass and low-pass signals .

Further information on the topic, numerous tasks and simulations can be found in the experiment „Analog Modulation Methods” of the practical course "Simulation of Digital Transmission Systems". This (former) LNT course at the TU Munich is based on

dem Windows Program AMV ⇒ Link refers to the ZIP version of the program and
the corresponding Praktikumsanleitung ⇒ Link refers to the PDF version; 86 pages in total.