View source for Signal Representation/Differences and Similarities of Low-Pass and Band-Pass Signals


{{Header
|Untermenü=Band-Pass Signals
|Vorherige Seite=The Convolution Theorem and Operation
|Nächste Seite=Analytical Signal and Its Spectral Function
}}

== # OVERVIEW OF THE FOURTH MAIN CHAPTER # ==
<br>
In the third main chapter &nbsp; &rArr; &nbsp; [[Signal_Representation|&raquo;Aperiodic Signals &ndash; Pulses&laquo;]]&nbsp; mostly&nbsp; "low-pass signals"&nbsp; were assumed &nbsp; &rArr; &nbsp; those signals whose spectral functions lie in the range around the frequency&nbsp; $f = 0$.&nbsp; 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&nbsp; $f_{\rm T}$.&nbsp; Such signals are called&nbsp; &raquo;'''band-pass signals'''&laquo;.

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.&nbsp; Besides there are some special features of band-pass signals,&nbsp; whose observance can lead to a simpler description.

This chapter contains in detail:
#The enumeration of&nbsp; &raquo;differences&nbsp; and&nbsp; similarities&laquo;&nbsp; of low-pass and band-pass signals,
#the&nbsp; &raquo;synthesis&nbsp; of band-pass signals&laquo;&nbsp; from the&nbsp; &raquo;equivalent low-pass signal&laquo;,
#the&nbsp; equivalent low-pass signal&nbsp;&nbsp; in the time and frequency domain,&nbsp; and finally
#the representation of the&nbsp; &raquo;analytical signal&laquo;&nbsp;&nbsp; and of the&nbsp; &raquo;equivalent low-pass signal&laquo;&nbsp; in the complex plane.




==Motivation of band-pass signals for Communications Engineering==
<br>
In the previous chapters of this book,&nbsp; only signals have been considered whose spectra lie in a narrow range around the frequency&nbsp; $f = 0$.&nbsp;  Examples are analog speech,&nbsp; music and image signals,&nbsp; which all &ndash; despite their different bandwidths &ndash; can be described as&nbsp; &raquo;low-pass signals&laquo;.

If you want to transmit such a low-pass signal to a spatially distant sink,&nbsp; the signal may have to be converted to another frequency position.&nbsp; 

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,&nbsp; because this band contains frequencies that are unfavorable for it.&nbsp; Only by a frequency shift by means of a so-called&nbsp; &raquo;modulator&laquo;&nbsp; a transmission is made possible.

*A single transmission channel can be used for the simultaneous transmission of several signals,&nbsp; if they are modulated with different carrier frequencies at the transmitting end.&nbsp; This method is called&nbsp; &nbsp; [[Modulation_Methods/Objectives_of_Modulation_and_Demodulation#Channel_bundling_.E2.80.93_Frequency_Division_Multiplexing|&raquo;Frequency Division Multiple Access&raquo;]]&nbsp; $\text{(FDMA)}$.

*The transmission quality can be improved compared to the simplest analog method&nbsp; [[Modulation_Methods/Zweiseitenband-Amplitudenmodulation|&raquo;Amplitude Modulation&laquo;]]&nbsp; $\text{(AM)}$&nbsp; at the expense of a larger bandwidth and thus a higher&nbsp; &raquo;signal-to-noise ratio&raquo;&nbsp; $\text{(SNR)}$&nbsp; can be achieved.&nbsp; Examples are the analog&nbsp;  [[Modulation_Methods/Frequenzmodulation_(FM)|&raquo;Frequency Modulation&laquo;]]&nbsp; $\text{(FM)}$&nbsp; and the digital &nbsp; [[Modulation_Methods/Pulse_Code_Modulation|&raquo;Pulse Code Modulation&laquo;]]&nbsp; $\text{(PCM)}$.


{{BlaueBox|TEXT=  
$\text{Remember:}$&nbsp; The transmitted signals of many transmission methods are&nbsp; &raquo;band-pass signals&laquo;.}}


{{GraueBox|TEXT=  
$\text{Example 1 : To classify signals with respect to&nbsp; &raquo;low-pass&laquo;&nbsp; and&nbsp; &raquo;band-pass&laquo;}$

'''(a)''' &nbsp; &nbsp; Speech and music are low-pass signals with a bandwidth of&nbsp; $\text{20 kHz}$&nbsp; $($at very good quality$)$.&nbsp; Since a radio transmission is only possible from&nbsp; $\approx \text{100 kHz}$,&nbsp; a conversion is needed to carrier frequencies between
*$\text{0.525 ... 1.61 MHz}$&nbsp; $($medium wave broadcasting,&nbsp; amplitude modulation,&nbsp; channel spacing&nbsp; $\text{9 kHz})$,

*$\text{87.5 ... 108 MHz}$&nbsp; $($radio on FM,&nbsp; frequency modulation,&nbsp; channel spacing&nbsp; $\text{300 kHz})$.


'''(b)''' &nbsp; &nbsp; TV image signals&nbsp; have a larger bandwidth,&nbsp; e.g.&nbsp; $\text{5 MHz}$.&nbsp; Here,&nbsp;  as well,&nbsp;  a frequency band shift occurs before the sound and image transmission due to carrier frequencies between
*$\text{41 ... 68 / 174 ... 230 MHz}$&nbsp; $($television,&nbsp; VHF band,&nbsp; channel spacing&nbsp; $\text{7 MHz})$,

*$\text{470 ... 850 MHz}$&nbsp; $($television,&nbsp; UHF band,&nbsp; channel spacing&nbsp; $\text{8 MHz})$.


'''(c)'''&nbsp; &nbsp; With&nbsp; &raquo;GSM mobile radio&laquo;&nbsp; the carrier frequencies in the D-band are&nbsp; $\text{900 MHz}$&nbsp; and in the E-band &nbsp; $\text{1800 MHz}$.  

'''(d)''' &nbsp; &nbsp; With&nbsp; &raquo;optical communication&laquo;&nbsp; the electrical signals are converted into light,&nbsp; i.e. to frequencies &nbsp; $\text{200 THz ... 350 THz}$&nbsp; &rArr; &nbsp; wavelengths:&nbsp; $\text{1.55 &micro;m ... 0.85 &micro;m}$.}}


==Properties of band-pass signals==
<br>
In this section &ndash; without claiming to be complete &ndash; some characteristics of band-pass signals&nbsp; $x_{\rm BP}(t)$&nbsp; are compiled and compared to low-pass signals&nbsp;  $x_{\rm TP}(t)$,&nbsp; because of&nbsp; "low-pass" &nbsp; &rArr; &nbsp; German:&nbsp; "Tiefpass" &nbsp; &rArr; &nbsp; "$\rm TP$".&nbsp; We start from the spectral functions&nbsp; $X_{\rm TP}(f)$&nbsp; and&nbsp; $X_{\rm BP}(f)$&nbsp; according to the following sketch.

Regarding the graphic is to be remarked:
[[File:P_ID679__Sig_T_4_1_S2a_neu.png|right|frame|Low-pass and band-pass spectrum]]

#The triangular shape of the displayed spectra is to be understood schematically and is only to mark the occupied frequency band.&nbsp; 
#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&nbsp; $x_{\rm TP}(t)$&nbsp; and&nbsp; $x_{\rm BP}(t)$&nbsp; are real.&nbsp; This means that according to the&nbsp;  [[Signal_Representation/Fourier_Transform_Theorems#Assignment_Theorem|&raquo;Assignment Theorem&laquo;]]&nbsp; the spectral functions&nbsp; $X_{\rm TP}(f)$&nbsp; and&nbsp; $X_{\rm BP}(f)$&nbsp;  each have an even real part and an odd imaginary part&nbsp; $($related to the frequency&nbsp; $f = 0)$.
#As bandwidth&nbsp; $B_{\rm TP}$&nbsp; or&nbsp; $B_{\rm BP}$&nbsp; for both low-pass and band-pass we refer to the occupied frequency band at positive frequencies&nbsp; $($in the graph: &nbsp; solid lines$)$.


{{GraueBox|TEXT=  
$\text{Example 2:}$&nbsp; An example with discrete spectral lines follows. 
[[File:P_ID698__Sig_T_4_1_S2b_neu.png|right|frame|Example of a low-pass  spectrum (left) and of a band-pass spectrum  (right)]]

The left graph shows the spectrum&nbsp; $Q(f)$&nbsp; of the source signal
 
:$$q(t) = 3\hspace{0.05cm}{\rm V} + 4\hspace{0.05cm}{\rm V} \cdot \cos (2 \pi \hspace{-0.05cm} \cdot\hspace{-0.05cm} 3\hspace{0.05cm}{\rm kHz} \cdot t) + 2\hspace{0.05cm}{\rm V} \cdot \sin (2 \pi\hspace{-0.05cm} \cdot\hspace{-0.05cm} 4\hspace{0.05cm}{\rm kHz} \cdot t). $$

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


&rArr; &nbsp; On the right you see the spectrum&nbsp; $S(f)$&nbsp; after&nbsp; &raquo;Single&ndash;sideband Amplitude Modulation&laquo;&nbsp; $\text{(SSB-AM)}$&nbsp; with carrier frequency $f_{\rm T} = 100 \,\text{kHz}$.&nbsp; 

A description of this special transmission system can be found in chapter&nbsp; [[Modulation_Methods/Hüllkurvendemodulation|&raquo;Envelope Demodulation&raquo;]]&nbsp; of the book&nbsp; &raquo;Modulation Methods&laquo;.
*According to this description,&nbsp; $q(t)$&nbsp; is uniquely a low-pass signal and&nbsp; $s(t)$&nbsp; is a band-pass signal.&nbsp; The bandwidths are&nbsp; $B_{\rm TP} = B_{\rm BP} = 4 \,\text{kHz}$.
*The signals&nbsp; $q(t)$&nbsp; and&nbsp; $s(t)$&nbsp; are real,&nbsp; since both&nbsp; $Q(f)$&nbsp; and&nbsp; $S(f)$&nbsp; have an even real and an odd imaginary part.
*If the DC component&nbsp; $(3 \,\text{V})$&nbsp; would be missing in the source signal,&nbsp; one would sensibly&nbsp; $q(t)$&nbsp; still call it&nbsp; "low-pass"&nbsp; with bandwidth&nbsp; $B_{\rm TP} = 4 \,\text{kHz}$.
*But without knowledge of the task,&nbsp; one could interpret&nbsp; $q(t)$&nbsp; then also as a&nbsp; "band-pass signal"&nbsp; with bandwidth&nbsp; $B_{\rm BP} = 1 \,\text{kHz}$.


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

==Description of a band-pass signal with low-pass signals==
<br>
We consider two low-pass spectra&nbsp; $X_1(f)$&nbsp; and&nbsp; $X_2(f)$&nbsp; with the bandwidths&nbsp; $B_1$&nbsp; and&nbsp; $B_2$&nbsp; corresponding to the left graph. 

[[File:P_ID684__Sig_T_4_1_S3a.png|right|frame|Generation of a band-pass spectrum from two low-pass spectra]]

You can see from this diagram:
*If&nbsp; $X_1(f)$&nbsp; and&nbsp; $X_2(f)$&nbsp; are identical up to a frequency&nbsp; $f_{12}$,&nbsp; the difference describes a band-pass spectrum with bandwidth&nbsp; $B_{\rm BP} = B_1 - f_{12}$.&nbsp; 

*According to the graph 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,&nbsp; the time function associated with the band-pass spectrum&nbsp; $X_{\rm BP}(f)$&nbsp; is valid:
:$$x_{\rm BP}(t) = x_1(t) - x_2(t).$$
*It generally follows from the Fourier transform that the integral over the time function is equal to the spectral value at&nbsp; $f = 0$.&nbsp;

* Consequently,&nbsp; this integral is always zero for every band-pass signal:
:$$\int_{- \infty}^{+\infty}x_{\rm BP}(t)\hspace{0.1cm}{\rm
d}t = X_{\rm BP}(f \hspace{-0.1cm}= \hspace{-0.1cm} 0) =0.$$

{{GraueBox|TEXT=  
$\text{Example 3:}$&nbsp;
The red curves in the two graphs show the band-pass spectrum&nbsp; $X_{\rm BP}(f)$&nbsp; and the corresponding time function
 
:$$x_{\rm BP}(t) = 10\hspace{0.05cm}{\rm
V} \cdot {\rm sinc} ( 10 \hspace{0.05cm}{\rm kHz} \cdot t)
\cdot {\rm sinc} ( 2 \hspace{0.05cm}{\rm kHz} \cdot t) -
2\hspace{0.05cm}{\rm V} \cdot {\rm sinc} ( 2
\hspace{0.05cm}{\rm kHz} \cdot t).$$

[[File:EN_Sig_T_4_1_S3a.png|right|frame|Low-pass and band-pass spectrum and their signals]]

Also shown are two low-pass spectra and signals.&nbsp; You can see from these diagrams:
*The blue-dotted curve in the left graph represents the trapezoidal spectrum&nbsp; $X_1(f)$&nbsp; where the equivalent bandwidth is&nbsp; $\Delta f_1= 10 \,\text{kHz}$&nbsp; and the rolloff factor&nbsp; $r_1 = 0.2$.

*The blue-dotted curve in the right graphic shows the corresponding low-pass signal&nbsp; $x_1(t)$.&nbsp; The signal value at&nbsp; $t = 0$&nbsp; corresponds to the blue trapezoidal area of the spectrum&nbsp; $X_1(f)$: 
:$$x_1(t = 0) = 10 \,\text{V}.$$
*The green curve applies to the rectangular spectrum&nbsp; $X_2(f)$&nbsp; with the equivalent bandwidth&nbsp; $\Delta f_2= 2 \,\text{kHz}$.&nbsp; The corresponding time signal&nbsp; $x_2(t)$&nbsp; is&nbsp; $\rm sinc$-shaped.&nbsp; It holds: 
:$$x_2(t = 0) = 2 \,\text{V}.$$
*The red curve for the band-pass signal is the difference between the blue and green curve,&nbsp; both on the left and the right.&nbsp; Accordingly:
:$$x_{\rm BP}(t = 0)  = x_1(t = 0) - x_2(t = 0) = 8 \,\text{V},$$
:$$\Rightarrow \ \int_{- \infty}^{+\infty}x_{\rm BP}(t)\hspace{0.1cm}{\rm
d}t = X_{\rm BP}(f \hspace{-0.1cm}= \hspace{-0.1cm} 0) =0.$$}}


==Band-pass signal synthesis  from  the equivalent low-pass signal==
<br>
We consider a low-pass signal&nbsp; $x_{\rm TP}(t)$&nbsp; with spectrum&nbsp; $X_{\rm TP}(f)$&nbsp; according to the left sketch. 

[[File:P_ID2724__Sig_T_4_1_S4a.png|center|frame|Band-pass spectrum is obtained by shifting a low-pass spectrum on both sides]]

*If this signal is multiplied by a&nbsp; $($dimensionless$)$&nbsp; &raquo;harmonic oscillation&laquo;

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

:the convolution theorem for the spectrum of the signal yields&nbsp; $x_{\rm BP}(t) = x_{\rm TP}(t) \cdot z(t)$:
 
:$$X_{\rm BP}(f) =  X_{\rm TP}(f)\star Z(f) = {1}/{2}\cdot X_{\rm
TP} (f - f_{\rm T})+ {1}/{2}\cdot X_{\rm TP}(f + f_{\rm T}).$$

*Here it is considered that the&nbsp; [[Signal_Representation/The_Convolution_Theorem_and_Operation|&raquo;convolution&laquo;]]&nbsp; of the spectral function&nbsp; $X_{\rm TP}(f)$&nbsp; with the frequency-shifted Dirac delta&nbsp; $\delta (f - f_\rm {T})$&nbsp; yields the same spectral function &nbsp; $X_{\rm TP}(f-f_\rm {T})$&nbsp; shifted to the right by $f_\rm {T}$&nbsp;.

*From spectral function on the right 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&nbsp; $x_{\rm BP}(t)$&nbsp;is  given by the magnitude of&nbsp; $|x_{\rm TP}(t)|$.

*For example,&nbsp; this principle is applied to the&nbsp; [[Modulation_Methods/Zweiseitenband-Amplitudenmodulation|&raquo;amplitude modulation without carrier&laquo;]],&nbsp; which is discussed thoroughly in the book&raquo; &raquo;Modulation Methods&laquo;.


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


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

[[File:P_ID687__Sig_T_4_1_S4b.png|right|frame|DSB-AM signal with two different carrier frequencies]]

:$$x_{\rm TP}(t) = 0.26\cdot {\cos} ( \omega_1 \hspace{0.05cm} t + 20^{ \circ}) \hspace{0.18cm}+ 0.54\cdot {\cos} ( \omega_2 \hspace{0.05cm} t - 180^{ \circ}) +  $$
:$$\hspace{1.5cm}+ \  0.30\cdot {\cos} ( \omega_3 \hspace{0.05cm} t +
120^{ \circ}) +0.14\cdot {\cos} ( \omega_4 \hspace{0.05cm} t -40^{
\circ}).$$

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

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


The topic of this chapter is covered in the&nbsp; (German language) learning video&nbsp;  [[Eigenschaften_von_Tiefpass-_und_Bandpasssignalen_(Lernvideo)|&raquo;Eigenschaften von Tiefpass- und Bandpass-Signalen&laquo;]]&nbsp; &rArr; &nbsp;"Properties of low-pass and band-pass signals".}} 




==Exercises for the Chapter==
<br>
[[Aufgaben:Exercise 4.1: Low-Pass and Band-Pass Signals|Exercise 4.1: Low-Pass and Band-Pass Signals]]

[[Aufgaben:Exercise 4.1Z: High-Pass System|Exercise 4.1Z: High-Pass System]]

[[Aufgaben:Exercise 4.2: Rectangular-Shaped Spectra|Exercise 4.2: Rectangular Spectra]]

[[Aufgaben:Exercise 4.2Z: Multiplication with a  Sine Signal|Exercise 4.2Z: Multiplication with a  Sine Signal]]


 
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