Signal Representation/Spectrum Analysis - Revision history

Guenter at 15:02, 27 June 2023

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Guenter at 17:50, 10 December 2022

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Hwang at 18:09, 8 December 2022

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Hwang at 18:05, 8 December 2022

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Hwang at 18:01, 8 December 2022

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Hwang at 18:00, 8 December 2022

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 {{BlaueBox|TEXT=
 $\text{Definition:}$&nbsp;
-The term&nbsp; $\text{spectral leakage effect}$&nbsp; is used to describe the distortion of the spectrum of a periodic and thus temporally unlimited signal due to the implicit time limit of the Discrete Fourier Transform (DFT).&nbsp; This means that, for example, a spectrum analyzer
+The term&nbsp; &raquo;'''spectral leakage effect'''&laquo;&nbsp; is used to describe the distortion of the spectrum of a periodic and thus temporally unlimited signal due to the implicit time limit of the&nbsp; &raquo;Discrete Fourier Transform&laquo;&nbsp; $\rm (DFT)$.&nbsp; This means that,&nbsp; for example,&nbsp; a spectrum analyzer
-* fake frequency components that are not present in the time signal, and/or
+* fake frequency components that are not present in the time signal,&nbsp; and/or
 *actually existing spectral components are hidden by side lobes.}}
-The following&nbsp; $\text{Example 1}$&nbsp; will show that for a periodic signal the application of the&nbsp; [[Signal_Representation/Discrete_Fourier_Transform_(DFT)|$\text{Discrete  Fourier Transform}$]]&nbsp; $\text{(DFT)}$&nbsp;  is not useful without additional measures.&nbsp; The quality of the spectral analysis &ndash; i.e. the correctness of the spectrum found &ndash; is mainly determined here by the (more or less successful) adaptation of the DFT parameters to the signal parameters at hand.
+The following&nbsp; $\text{Example 1}$&nbsp; will show that for a periodic signal the application of the&nbsp; [[Signal_Representation/Discrete_Fourier_Transform_(DFT)|&raquo;Discrete  Fourier Transform&laquo;]]&nbsp; $\text{(DFT)}$&nbsp;  is not useful without additional measures.&nbsp; The quality of the spectral analysis &ndash; i.e. the correctness of the spectrum found &ndash; is mainly determined here by the&nbsp; $($more or less successful$)$&nbsp; adaptation of the DFT parameters to the signal parameters at hand.
-*If the period&nbsp; $T_0$&nbsp; of the signal is known, the duration&nbsp; $T_{\rm P}$&nbsp; of the signal section used for the DFT should be an integer multiple of&nbsp; $T_0$.&nbsp; However, the task of spectral analysis is precisely to find arbitrary signal components, so that the knowledge of&nbsp; $T_0$&nbsp; cannot generally be assumed.
+#If the period&nbsp; $T_0$&nbsp; of the signal is known,&nbsp; the duration&nbsp; $T_{\rm P}$&nbsp; of the signal section used for the DFT should be an integer multiple of&nbsp; $T_0$.&nbsp;
-*A measure to improve the spectral analysis is the windowing with a "suitable" time function&nbsp; $w(t)$.&nbsp; The product signal&nbsp; $x(t) \cdot w(t)$ is then analyzed.
+#However,&nbsp; the task of spectral analysis is precisely to find arbitrary signal components,&nbsp; so that the knowledge of&nbsp; $T_0$&nbsp; cannot generally be assumed.
-*A large number of such window functions&nbsp; $w(t)$&nbsp; are known from the literature, which lead to good or less satisfactory results depending on the task.
+#A measure to improve the spectral analysis is the windowing with a&nbsp; suitable&nbsp; time function&nbsp; $w(t)$.&nbsp; The product signal&nbsp; $x(t) \cdot w(t)$&nbsp; is then analyzed.
+#A large number of such window functions&nbsp; $w(t)$&nbsp; are known from the literature,&nbsp; which lead to good or less satisfactory results depending on the task.
-So much up front:&nbsp; '''There is no&nbsp; "best"&nbsp; window function for all applications'''.
+In the next sections the spectral leakage effect will be illustrated by examples and the advantages and disadvantages of the different window functions will be discussed.&nbsp; So much up front:&nbsp; '''There is no&nbsp; "best"&nbsp; window function for all applications'''.
 {{GraueBox|TEXT=
@@ Line 31: / Line 31: @@
 [[File:P_ID1160__Sig_T_5_4_S1_neu.png|right|frame|Example of applying  spectral analysis]]
-On the right the frequency discrete spectrum&nbsp; $\vert D(\mu) \vert$&nbsp; after a DFT with&nbsp; $N = 32$&nbsp; samples is shown in logarithmic form (in dB) , from which the further DFT parameters result as follows:
+&rArr; &nbsp; On the right the frequency discrete spectrum&nbsp; $\vert D(\mu) \vert$&nbsp; after a DFT with&nbsp; $N = 32$&nbsp; samples is shown in logarithmic form (in dB),&nbsp; from which the further DFT parameters result as follows:
 *Duration of the time segment: &nbsp; $T_{\rm P} = 32 \,{\rm &micro; s}$,
 *gridding of the frequency axis: &nbsp; $f_{\rm A} = 31.25 \,\text{ kHz}$.
-Since the interval width&nbsp; $T_{\rm P}$&nbsp; captures an integer multiple of the period&nbsp; $T_0$&nbsp;, the DFT delivers the correct result. The two Dirac delta functions lie exactly at&nbsp; $\pm4 \cdot f_{\rm A}$.
+Since the interval width&nbsp; $T_{\rm P}$&nbsp; captures an integer multiple of period&nbsp; $T_0$,&nbsp; the DFT delivers the correct result.&nbsp; The two Dirac delta functions lie exactly at&nbsp; $\pm4 \cdot f_{\rm A}$.
-If one measures an oscillation of frequency&nbsp; $f_0 = 109.375\,\text{ kHz}$ &nbsp; &rArr; &nbsp; period $T_0 = 9.14 \,{\rm &micro; s}$&nbsp; corresponding to the graph below&nbsp; '''(b)''', significant distortions of the spectrum occur.
+*Since now&nbsp; $T_{\rm P}/T_0 = 3.5$&nbsp; is no longer an integer,&nbsp; the periodic continuation of the time section causes phase jumps,&nbsp; in our example by&nbsp; $\pi$.
-*Since now&nbsp; $T_{\rm P}/T_0 = 3.5$&nbsp; is no longer an integer, the periodic continuation of the time section causes phase jumps, in our example by&nbsp; $\pi$.
+*The spectral range now no longer consists of two Dirac delta functions as in the sketch&nbsp; '''(a)''',&nbsp; but of an approximately continuous frequency function with the maximum near the actual signal frequency and a series of further parts,&nbsp; which are called&nbsp; &raquo;side lobes&laquo;.}}

@@ Line 36: / Line 36: @@
-Since the interval width&nbsp; $T_{\rm P}$&nbsp; captures an integer multiple of the period&nbsp; $T_0$&nbsp;, the DFT delivers the correct result. The two Dirac '''KORREKTUR: delta?''' functions lie exactly at&nbsp; $\pm4 \cdot f_{\rm A}$.
+Since the interval width&nbsp; $T_{\rm P}$&nbsp; captures an integer multiple of the period&nbsp; $T_0$&nbsp;, the DFT delivers the correct result. The two Dirac delta functions lie exactly at&nbsp; $\pm4 \cdot f_{\rm A}$.
 If one measures an oscillation of frequency&nbsp; $f_0 = 109.375\,\text{ kHz}$ &nbsp; &rArr; &nbsp; period $T_0 = 9.14 \,{\rm &micro; s}$&nbsp; corresponding to the graph below&nbsp; '''(b)''', significant distortions of the spectrum occur.
 *Since now&nbsp; $T_{\rm P}/T_0 = 3.5$&nbsp; is no longer an integer, the periodic continuation of the time section causes phase jumps, in our example by&nbsp; $\pi$.
-*The spectral range now no longer consists of two Dirac '''KORREKTUR: delta?''' functions as in the sketch&nbsp; '''(a)''', but of an approximately "continuous" frequency function with the maximum near the actual signal frequency and a series of further parts, which are called&nbsp; "side lobes".}}
+*The spectral range now no longer consists of two Dirac delta functions as in the sketch&nbsp; '''(a)''', but of an approximately "continuous" frequency function with the maximum near the actual signal frequency and a series of further parts, which are called&nbsp; "side lobes".}}

@@ Line 146: / Line 146: @@
   \frac{\int_{-\infty}^{\infty}|W(f)|^2\hspace{0.05cm}{\rm d}f}{f_{\rm A} \cdot |W(f=0)|^2} \hspace{0.05cm}.$$
-From the&nbsp; [[Signal_Representation/Spectrum_Analysis#G.C3.BCtekriterien_von_Fensterfunktionen|"table in the last section"]]&nbsp; it can be seen that for the window functions considered&nbsp; $V_{\rm P}$&nbsp; takes values between&nbsp; $\text{3 dB}$&nbsp; and&nbsp; $\text{4 dB}$,&nbsp; where window functions with&nbsp; $V_{\rm P} > 3. 7 \,\text{dB}$&nbsp; (Rectangle, Blackman-Harris, Kaiser-Bessel)&nbsp; should not be used.&nbsp; However, it is precisely these that are best with regard to the main-to-side lobe distance.
+From the&nbsp; [[Signal_Representation/Spectrum_Analysis#Quality_criteria_of_window_functions|"table in the last section"]]&nbsp; it can be seen that for the window functions considered&nbsp; $V_{\rm P}$&nbsp; takes values between&nbsp; $\text{3 dB}$&nbsp; and&nbsp; $\text{4 dB}$,&nbsp; where window functions with&nbsp; $V_{\rm P} > 3. 7 \,\text{dB}$&nbsp; (Rectangle, Blackman-Harris, Kaiser-Bessel)&nbsp; should not be used.&nbsp; However, it is precisely these that are best with regard to the main-to-side lobe distance.
 The two proportions are to be interpreted as follows:

@@ Line 132: / Line 132: @@
 [[File:EN_Sig_T_5_4_S4b.png|left|frame|Illustrating the "6dB bandwidth"]]
-*The&nbsp; &raquo;'''6 dB bandwidth'''&laquo;, which can be read from the logarithmised spectral function, is an important measure of the frequency resolving power.&nbsp; Two spectral components present in the signal at&nbsp; $f_1$&nbsp; and&nbsp; $f_2$&nbsp; can only be resolved if the difference&nbsp; $f_2 - f_1$&nbsp; is greater than the&nbsp; "6dB bandwidth" of the window function used (see left graph).
+*The&nbsp; &raquo;'''6 dB bandwidth'''&laquo;, which can be read from the logarithmized spectral function, is an important measure of the frequency resolving power.&nbsp; Two spectral components present in the signal at&nbsp; $f_1$&nbsp; and&nbsp; $f_2$&nbsp; can only be resolved if the difference&nbsp; $f_2 - f_1$&nbsp; is greater than the&nbsp; "6dB bandwidth" of the window function used (see left graph).
 *The&nbsp; &raquo;'''window area'''&laquo;&nbsp; of the function&nbsp; $w(t)$&nbsp; at the same time gives the height&nbsp; $W(0)$&nbsp; in the spectral domain.&nbsp; For all windows except the rectangle, a window area smaller than&nbsp; $1$&nbsp; and thus an error in the amplitude of the DFT results due to the suppression of the outer samples, which, however, can be completely corrected if&nbsp; $w(t)$&nbsp; is known.
@@ Line 158: / Line 158: @@
 {{BlaueBox|TEXT=
 $\text{Conclusion:}$&nbsp;
-The results of this section can be summarised as follows:
+The results of this section can be summarized as follows:
 *'''An ideal window function does not exist'''.&nbsp; Depending on the task&nbsp; (good amplitude or frequency resolution)&nbsp; different windows provide the best result in each case.&nbsp; It is therefore recommended that one always uses several window functions for spectral analysis or at least one window function with different parameters.
 *A workable compromise with regard to all criteria is the&nbsp; &raquo;'''Hamming window'''&laquo;, which only gives one unfavourable value&nbsp; $($"side lobe drop"&nbsp; only&nbsp; 6 dB per octave$)$&nbsp;.

@@ Line 89: / Line 89: @@
 \hspace{0.2cm}W(f) ={1}/({2f_{\rm A}})\cdot {\rm sinc}^2(
 {f}/({2f_{\rm A}}))\hspace{0.05cm}.$$
-*Due to the lower weighting of the edge regions, which are particularly problematic with unbounded signals, the (logarithmically drawn) spectrum&nbsp; $W(f)$&nbsp; has lower side lobes than the&nbsp; $\rm si$&ndash;function in the upper graph, which leads to lower leakage components.
+*Due to the lower weighting of the edge regions, which are particularly problematic with unbounded signals, the (logarithmically drawn) spectrum&nbsp; $W(f)$&nbsp; has lower side lobes than the&nbsp; $\rm sinc$ function in the upper graph, which leads to lower leakage components.
 *The better suppression of the side lobes, however, comes at the cost of a noticeable reduction and broadening of the main lobe, limiting the resolving power of the Bartlett window compared to rectangular windowing.

@@ Line 52: / Line 52: @@
 First consider the upper graph&nbsp; '''(a)'''&nbsp; for the&nbsp; &raquo;'''rectangular window'''&laquo;.
 *The time limit implicit in the DFT corresponds to the multiplication of the signal&nbsp; $x(t)$&nbsp; by a rectangular window function&nbsp; $w(t)$&nbsp; of height&nbsp; $1$&nbsp; and duration&nbsp; $T_{\rm P}$.
-*The upper left graph shows the discrete-time representation of this rectangular function with the normalised time variable&nbsp; $\nu= t/T_{\rm A}$:
+*The upper left graph shows the discrete-time representation of this rectangular function with the normalized time variable&nbsp; $\nu= t/T_{\rm A}$:
 :$${w} (\nu)   = \left\{ \begin{array}{c} 1 \\
@@ Line 69: / Line 69: @@
 *Otherwise, convolution with&nbsp; $W(f)$&nbsp; leads to distortions, since the zeros of the&nbsp; $\rm sinc$ function no longer fit the discrete values of the input spectrum.
 <br clear=all>
-The discontinuities in the time domain caused by limitation and periodic continuation are reduced if, instead of the constant one weighting by the rectangle, the two edge areas of the window are weighted weaker than the centre.
+The discontinuities in the time domain caused by limitation and periodic continuation are reduced if, instead of the constant one weighting by the rectangle, the two edge areas of the window are weighted weaker than the center.
 Consider the graph&nbsp; '''(b)'''&nbsp; for the&nbsp; &raquo;'''Bartlett window'''&laquo; - also called&nbsp; "triangular window":
@@ Line 140: / Line 140: @@
 ==Maximum process loss==
 <br>
-This combined quality criterion considers the&nbsp; &raquo;'''maximum scaling error'''&laquo;&nbsp; as well as the (normalised)&nbsp; &raquo;'''equivalent noise bandwidth'''&laquo;.&nbsp; The maximum process loss is usually given in&nbsp; $\text{dB}$&nbsp; and should be rather small according to its name:
+This combined quality criterion considers the&nbsp; &raquo;'''maximum scaling error'''&laquo;&nbsp; as well as the (normalized)&nbsp; &raquo;'''equivalent noise bandwidth'''&laquo;.&nbsp; The maximum process loss is usually given in&nbsp; $\text{dB}$&nbsp; and should be rather small according to its name:
 :$$10 \cdot {\rm lg}\hspace{0.15cm}V_{\rm P}\hspace{0.15cm}{\rm (in}\hspace{0.15cm}{\rm dB)}= 20 \cdot {\rm lg}\hspace{0.15cm}

@@ Line 64: / Line 64: @@
 *From the multiplication&nbsp; $y(t) = x(t) \cdot w(t)$&nbsp; of the signal&nbsp; $x(t)$&nbsp; to be analyzed and the window function&nbsp; $w(t)$&nbsp; follows for the spectral function&nbsp; $Y(f) = X(f) \ast W(f)$, where for the rectangular window with&nbsp; $f_{\rm A} = 1/T_{\rm P}$&nbsp; and&nbsp; ${\rm si}(x) = \sin(x)/x = {\rm sinc}(x/\pi) $&nbsp; holds:
-:$$W(f) = T_{\rm P} \cdot {\rm sinc}(f \cdot T_{\rm P}) = {1}/{f_{\rm A}}\cdot {\rm si}(\pi \cdot {f}/{f_{\rm A}})\hspace{0.05cm}.$$
+:$$W(f) = T_{\rm P} \cdot {\rm sinc}(f \cdot T_{\rm P}) = {1}/{f_{\rm A}}\cdot {\rm sinc}({f}/{f_{\rm A}})\hspace{0.05cm}.$$
 *The function&nbsp; $W(f)$&nbsp; is shown logarithmically in the upper right graph.&nbsp; If all spectral components of&nbsp; $x(t)$&nbsp; lie in the frequency grid&nbsp; $\mu \cdot f_{\rm A}$, the discrete frequency spectral values&nbsp; $D(\mu )$&nbsp; remain unchanged by the convolution with&nbsp; $W(f)$.