Signal Representation/Possible Errors when using DFT - Revision history

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 {{BlaueBox|TEXT=
 $\text{Definition:}$&nbsp;
-A quality criteria that takes both error types into account is the&nbsp; &raquo;'''mean square error'''&laquo;&nbsp; $($German:&nbsp; mittlerer quadratischer Fehler &nbsp; &rArr; &nbsp;$\rm (MQF)$:
+A quality criteria that takes both error types into account is the&nbsp; &raquo;'''mean square error'''&laquo;&nbsp; $($German:&nbsp; "mittlerer quadratischer Fehler" &nbsp; &rArr; &nbsp;$\rm MQF)$:
 :$${\rm MQF} =  \frac{1}{N}\cdot \sum_{\mu = 0 }^{N-1}

@@ Line 36: / Line 36: @@
 We refer you already here to the following&nbsp; $($German language$)$&nbsp; learning video,&nbsp; which clarifies the content of this chapter:<br>&nbsp; &nbsp; [[Fehlermöglichkeiten_bei_Anwendung_der_DFT_(Lernvideo)|&raquo;Fehlermöglichkeiten bei Anwendung der DFT&laquo;]] &nbsp; &rArr; &nbsp; "Possible errors when using the DFT".
 {{GraueBox|TEXT=
 $\text{Example 1:}$&nbsp;
-As an example, we consider a Gaussian pulse with the equivalent pulse duration&nbsp; $\Delta t = T$, where&nbsp; $T$&nbsp; is simultaneously used as a normalization parameter:
+As an example,&nbsp; we consider a Gaussian pulse with equivalent pulse duration&nbsp; $\Delta t = T$,&nbsp; where&nbsp; $T$&nbsp; is simultaneously used as a normalization parameter:
-[[File:EN_Sig_T_5_1_S1.png|right|frame|Quasi error-free DFT  with &nbsp; $N = 16$; &nbsp; &nbsp; ${\rm MQF}$: &nbsp; Mean square error]]
+[[File:EN_Sig_T_5_1_S1.png|right|frame|Quasi error-free DFT  with &nbsp; $N = 16$; &nbsp; &nbsp; ${\rm MQF}$: &nbsp; mean square error &nbsp; &nbsp; <u>Note:</u><br>The given&nbsp; $\text{MQF}$ values apply to a&nbsp; 16 bit processor.&nbsp; For a 32 bit processor&nbsp; $($smaller quantization errors of the computer$)$&nbsp; $\text{MQF}$&nbsp; would be even smaller,&nbsp; but never zero.]]
 :$$x(t) = {\rm e}^{- \pi (t/T)^2} \hspace{0.05cm}.$$
-The Gaussian pulse is very suitable for the application of DFT due to its fast, exponential decay in both the time and frequency domain.
+The Gaussian pulse is very suitable for the application of the  DFT due to its fast&nbsp; $($exponential$)$&nbsp; decay in both time and frequency domain.
 The graph below shows the DFT result
 *for&nbsp; $N = 16$&nbsp; and
 *$T_{\rm A}/T = 0.25$ &nbsp; &rArr; &nbsp; $f_{\rm A} \cdot T = 0.25$ &nbsp; &rArr; &nbsp; $T_{\rm P}/T = 4$.
-<br clear=all>
 The following should be noted about this plot:
-*The considered samples of&nbsp; $x(t)$&nbsp; are in the range&nbsp; $\vert t/T \vert≤ 2$.&nbsp; Since&nbsp; $x(\pm 2T)$&nbsp; is very small, periodization in the time domain with&nbsp; $T_{\rm P}/T = N \cdot T_{\rm A}/T = 4$&nbsp; does not lead to serious errors.
+#The considered samples of&nbsp; $x(t)$&nbsp; are in the range&nbsp; $\vert t/T \vert≤ 2$.&nbsp; Since&nbsp; $x(\pm 2T)$&nbsp; is very small,&nbsp; periodization in time domain with&nbsp; $T_{\rm P}/T = N \cdot T_{\rm A}/T = 4$&nbsp; does not lead to serious errors.
-*With&nbsp; $f_{\rm A} \cdot T = 0.25$&nbsp; and&nbsp; $N = 16$&nbsp; the (normalized) DFT parameter is&nbsp; $f_{\rm P} \cdot T = 4$.&nbsp; The discrete spectral lines thus lie in the range&nbsp; $-2/T ≤ f < +2/T$.
+#With&nbsp; $f_{\rm A} \cdot T = 0.25$&nbsp; and&nbsp; $N = 16$&nbsp; the&nbsp; $($normalized$)$&nbsp; DFT parameter is&nbsp; $f_{\rm P} \cdot T = 4$.&nbsp; The discrete spectral lines thus lie in the range&nbsp; $-2/T ≤ f < +2/T$.
-*The mean squared error is relatively small&nbsp; $\text{(MQF} \approx 10^{-12})$, which is due to the favourable choice of&nbsp; $f_{\rm A} \cdot T = 0.25$&nbsp; $($for a given&nbsp; $N = 16)$&nbsp;.
+#The mean squared error is relatively small&nbsp; $\text{(MQF} \approx 10^{-12})$,&nbsp; which is due to the favourable choice of&nbsp; $f_{\rm A} \cdot T = 0.25$&nbsp; $($for the given&nbsp; $N = 16)$&nbsp;.
-*The DFT accuracy can be improved by increasing&nbsp; $N$&nbsp;:
+#The DFT accuracy can be improved by increasing&nbsp; $N$&nbsp;:
-:*For&nbsp; $N = 1024$&nbsp; the minimum value&nbsp; $\text{MQF} \approx 8 \cdot 10^{-17}$&nbsp;  is obtained, if&nbsp; $f_{\rm A} \cdot T = 0.125$&nbsp; is chosen. The following then applies to the other DFT parameters:
+#For&nbsp; $N = 1024$&nbsp; the minimum value&nbsp; $\text{MQF} \approx 8 \cdot 10^{-17}$&nbsp;  is obtained,&nbsp; if&nbsp; $f_{\rm A} \cdot T = 0.125$&nbsp; is chosen.
-:: $$f_{\rm P} \cdot T = 128, \hspace{0.5cm}T_{\rm A}/T = 1/128, \hspace{0.5cm} T_{\rm P}/T = N \cdot T_{\rm A}/T= 8.$$
+#The following then applies to the other DFT parameters:
-:*The given&nbsp; $\text{MQF}$ values apply to a&nbsp; 16 bit processor.&nbsp; For a 32 bit processor&nbsp; $($meaning:&nbsp; smaller quantization errors of the computer$)$&nbsp; $\text{MQF}$&nbsp; would be even smaller, but never zero. }}
+:: $$f_{\rm P} \cdot T = 128, \hspace{0.5cm}T_{\rm A}/T = 1/128, \hspace{0.5cm} T_{\rm P}/T = N \cdot T_{\rm A}/T= 8.$$ }}

@@ Line 8: / Line 8: @@
 ==The mean square error as a quality criteria==
 <br>
-In the following, we briefly discuss some error possibilities when applying the DFT, whereby we restrict ourselves to the transformation from the time to the frequency domain.&nbsp; Even in its samples, the spectrum&nbsp; $D(\mu )/f_{\rm A}$&nbsp; determined via the DFT will generally differ from the actual spectrum&nbsp; $X(\mu \cdot f_{\rm A})$&nbsp; due to two processes:
+In the following,&nbsp; we briefly discuss some error possibilities when applying the DFT,&nbsp; whereby we restrict ourselves to the transformation from the time to the frequency domain.&nbsp; Even in its samples,&nbsp; the spectrum&nbsp; $D(\mu )/f_{\rm A}$&nbsp; determined via the DFT will generally differ from the actual spectrum&nbsp; $X(\mu \cdot f_{\rm A})$&nbsp; due to two processes:
-*the&nbsp; $\text{sampling}$, that is, the reduction of information about&nbsp; $x(t)$&nbsp; to&nbsp; $N$&nbsp; numerical values,
+*the&nbsp; &raquo;sampling&laquo;,&nbsp; that is,&nbsp; the reduction of information about&nbsp; $x(t)$&nbsp; to&nbsp; $N$&nbsp; numerical values,
-*the&nbsp; $\text{windowing}$ that may falsely limit the signal signal&nbsp; $x(t)$.
 {{BlaueBox|TEXT=
 $\text{Definition:}$&nbsp;
-A quality criteria that takes both error types into account is the&nbsp; $\text{mean square error}$&nbsp; $($German:&nbsp; mittlerer quadratischer Fehler &nbsp; &rArr; &nbsp;$\rm (MQF)$:
+A quality criteria that takes both error types into account is the&nbsp; &raquo;'''mean square error'''&laquo;&nbsp; $($German:&nbsp; mittlerer quadratischer Fehler &nbsp; &rArr; &nbsp;$\rm (MQF)$:
 :$${\rm MQF} =  \frac{1}{N}\cdot \sum_{\mu = 0 }^{N-1}
   \left\vert X(\mu \cdot f_{\rm A})-\frac{D(\mu)}{f_{\rm A} }\right \vert^2 \hspace{0.05cm}.$$
-It is always&nbsp; ${\rm MQF} \ne 0$, since with finite&nbsp; $N$&nbsp; degradation due to sampling and due to windowing cannot be made zero at the same time.}}
+It is always&nbsp; ${\rm MQF} \ne 0$,&nbsp; since with finite&nbsp; $N$&nbsp; the degradation due to sampling and due to windowing cannot be made zero at the same time.}}
 The magnitude of this evaluation quantity&nbsp; ${\rm MQF}$&nbsp; depends on the following parameters:
-*The properties of the present signal&nbsp; $x(t)$&nbsp; and of its spectrum&nbsp; $X(f)$,
+#The properties of the present signal&nbsp; $x(t)$&nbsp; and of its spectrum&nbsp; $X(f)$,
-*the DFT parameter&nbsp; $N$;&nbsp; the larger&nbsp; $N$&nbsp; is chosen, the smaller&nbsp; ${\rm MQF}$ becomes,
+#the DFT parameter&nbsp; $N$; &nbsp; the larger&nbsp; $N$&nbsp; is chosen,&nbsp; the smaller&nbsp; ${\rm MQF}$&nbsp; becomes,
-*one of the four further DFT parameters, for example&nbsp; $f_{\rm A}$.
+#one of the four further DFT parameters,&nbsp; e.g.&nbsp; $f_{\rm A}$.
-For a given&nbsp; $N$&nbsp; the other DFT parameters are determined via the equations&nbsp; $f_{\rm P} = N \cdot f_{\rm A}$,&nbsp; $T_{\rm P} = 1/f_{\rm A}$&nbsp; and&nbsp; $T_{\rm A} = T_{\rm P}/N$.
+For a given&nbsp; $N$&nbsp; the other DFT parameters are determined via the equations&nbsp;
-We refer you already here to the following (German language) learning video, which clarifies the content of this chapter:<br>&nbsp; &nbsp; [[Fehlermöglichkeiten_bei_Anwendung_der_DFT_(Lernvideo)|"Fehlermöglichkeiten bei Anwendung der DFT"]] &nbsp; &rArr; &nbsp; "Possible errors when using the DFT".
+We refer you already here to the following&nbsp; $($German language$)$&nbsp; learning video,&nbsp; which clarifies the content of this chapter:<br>&nbsp; &nbsp; [[Fehlermöglichkeiten_bei_Anwendung_der_DFT_(Lernvideo)|&raquo;Fehlermöglichkeiten bei Anwendung der DFT&laquo;]] &nbsp; &rArr; &nbsp; "Possible errors when using the DFT".
 {{GraueBox|TEXT=
 $\text{Example 1:}$&nbsp;
 As an example, we consider a Gaussian pulse with the equivalent pulse duration&nbsp; $\Delta t = T$, where&nbsp; $T$&nbsp; is simultaneously used as a normalization parameter:
 :$$x(t) = {\rm e}^{- \pi (t/T)^2} \hspace{0.05cm}.$$

@@ Line 62: / Line 62: @@
 *The windowing implicit in the DFT algorithm corresponds to the multiplication of the signal&nbsp; $x(t)$ by&nbsp; a rectangular function of height&nbsp; $1$&nbsp; and duration&nbsp; $T_{\rm P} = N \cdot T_{\rm A}$.
 *If the signal&nbsp; $x(t)$&nbsp; is not limited to the range&nbsp; $T_{\rm P}$&nbsp; the DFT result does not coincide with the actual spectrum&nbsp; $X(f)$&nbsp; but is obtained from it by convolution with the spectral function&nbsp; $T_{\rm P} \cdot \text{sinc}(fT_{\rm P})$,&nbsp; with&nbsp; $\text{si}(x) = \sin(x)/x=\text{sinc}(x/\pi)$ .
-*In the limiting case&nbsp; $T_{\rm P} \to \infty$, which for a given distance&nbsp; $T_{\rm A}$&nbsp; of the samples would also mean an infinitely large number&nbsp; $N$&nbsp; of interpolation points,&nbsp; $T_{\rm P} \cdot \text{sinc}(fT_{\rm P})$ degenerates&nbsp; to a Dirac function and the original spectrum&nbsp; $X(f)$&nbsp; would remain.
+*In the limiting case&nbsp; $T_{\rm P} \to \infty$, which for a given distance&nbsp; $T_{\rm A}$&nbsp; of the samples would also mean an infinitely large number&nbsp; $N$&nbsp; of interpolation points,&nbsp; $T_{\rm P} \cdot \text{sinc}(fT_{\rm P})$ degenerates&nbsp; to a Dirac delta function and the original spectrum&nbsp; $X(f)$&nbsp; would remain.
 *The DFT of an unlimited signal in time - for example a periodic signal - will always cause a truncation error which can only be kept within limits by special measures.&nbsp; This is discussed in more detail in the chapter&nbsp; [[Signal_Representation/Spectrum_Analysis|"Spectrum Analysis"]]&nbsp;.
 *For time-limited signals &nbsp; &rArr; &nbsp; "pulses",&nbsp; the truncation error can be avoided by choosing&nbsp; $T_{\rm P}$&nbsp; sufficiently large.&nbsp; By further enlarging the window into areas with&nbsp; $x(t) \equiv 0$&nbsp; no additional information gain results &nbsp; ⇒ &nbsp; $\text{MQF}$&nbsp; does not become smaller.

@@ Line 115: / Line 115: @@
 *However, aliasing now arises, because the DFT is derived from many Gaussian functions at distance&nbsp; $f_{\rm P}  \cdot T = 2$&nbsp; (thin dashed curves in the graph).
 *The DFT coefficients are falsified differently: &nbsp; The mean DFT coefficient&nbsp; $($for frequency&nbsp; $f = 0)$&nbsp; is almost correct, while the errors of the DFT coefficients increase significantly towards the edges.
-*In this example, the DFT coefficient for&nbsp; $f \cdot T = -1$&nbsp; is twice as large as it should be, since the Gaussian function with the centre at&nbsp; $f \cdot T = -2$&nbsp; gives the same contribution as the actual Gaussian function around&nbsp; $f \cdot T = 0$&nbsp; (see yellow background).
+*In this example, the DFT coefficient for&nbsp; $f \cdot T = -1$&nbsp; is twice as large as it should be, since the Gaussian function with the center at&nbsp; $f \cdot T = -2$&nbsp; gives the same contribution as the actual Gaussian function around&nbsp; $f \cdot T = 0$&nbsp; (see yellow background).
-Thus, here with&nbsp; $\text{MQF} \approx 2 \cdot 10^{-4}$&nbsp; an error value four times larger than that caused by the termination error in&nbsp; [[Signal_Representation/Possible_Errors_When_Using_DFT#DFT-Verf.C3.A4lschung_durch_Fensterung_.E2.80.93_Abbruchfehler|$\text{Example 2}$]].
+Thus, here with&nbsp; $\text{MQF} \approx 2 \cdot 10^{-4}$&nbsp; an error value four times larger than that caused by the termination error in&nbsp; [[Signal_Representation/Possible_Errors_when_using_DFT#DFT_falsification_due_to_windowing_.E2.80.93_Truncation_error|$\text{Example 2}$]].
 Again, we refer to the learning video (in German language):&nbsp; [[Fehlermöglichkeiten_bei_Anwendung_der_DFT_(Lernvideo)|"Fehlermöglichkeiten bei Anwendung der DFT"]] &nbsp; &rArr; &nbsp; "Possible errors when using the DFT". }}

@@ Line 102: / Line 102: @@
-[[File:EN_Sig_T_5_1_S3._neu.png|right|frame|Aliasing Error of a  DFT With&nbsp; $N = 16$; &nbsp; &nbsp; ${\rm MQF}$: &nbsp; Mean square error]]
+[[File:EN_Sig_T_5_1_S3._neu.png|right|frame|Aliasing error of a DFT with&nbsp; $N = 16$; &nbsp; &nbsp; ${\rm MQF}$: &nbsp; Mean square error]]
 {{GraueBox|TEXT=
 $\text{Example 3:}$&nbsp;
 Let the DFT parameters be&nbsp; $N = 16$&nbsp; and&nbsp; $f_{\rm A} \cdot T= 0.125$. Thus, for the other three DFT parameters we get:
-* $T_{\rm P}/T = 8 \hspace{0.8cm} \text{(Beispiel 1:} \ \ T_{\rm P}/T = 4)$,
+* $T_{\rm P}/T = 8 \hspace{0.8cm} \text{(Example 1:} \ \ T_{\rm P}/T = 4)$,
-* $f_{\rm P} \cdot T = 2 \hspace{0.75cm} \text{(Beispiel 1:} \ \ f_{\rm P} \cdot T = 4)$,
+* $f_{\rm P} \cdot T = 2 \hspace{0.75cm} \text{(Example 1:} \ \ f_{\rm P} \cdot T = 4)$,
-* $T_{\rm A}/T = 0.5\hspace{0.45cm} \text{(Beispiel 1:} \ \ T_{\rm A}/T = 0.25)$.
+* $T_{\rm A}/T = 0.5\hspace{0.45cm} \text{(Example 1:} \ \ T_{\rm A}/T = 0.25)$.

@@ Line 91: / Line 91: @@
 An unsuitable sampling of the time function&nbsp; $x(t)$&nbsp; can also significantly falsify the DFT result.&nbsp; This so-called&nbsp; $\text{aliasing error}$&nbsp; can be explained as follows:
-*Sampling&nbsp; $x(t)$&nbsp; at a distance&nbsp; $T_{\rm A}$&nbsp; causes a periodic continuation of the spectrum at multiples of the periodisation frequency&nbsp; $f_{\rm P} = 1/T_{\rm A}$.
+*Sampling&nbsp; $x(t)$&nbsp; at a distance&nbsp; $T_{\rm A}$&nbsp; causes a periodic continuation of the spectrum at multiples of the periodization frequency&nbsp; $f_{\rm P} = 1/T_{\rm A}$.
 *If&nbsp; $X(f)$&nbsp; has spectral components at&nbsp; $|f| > f_{\rm P}/2$, the sampling theorem is not fulfilled and overlaps of the shifted frequency components to be added occur.
 *Only with a band-limited signal the aliasing error can be avoided by suitable DFT parameters.&nbsp; In contrast, this error is unavoidable with time-limited signals &nbsp; &rArr; &nbsp; "pulses", since time-limited signals cannot be band-limited at the same time.

@@ Line 98: / Line 98: @@
 The following&nbsp; $\text{Example 3}$&nbsp; shows such an aliasing error due to wrongly chosen DFT parameters:
-*Compared to the&nbsp; "Comparison System"&nbsp; according to&nbsp; [[Signal_Representation/Possible_Errors_when_using_DFT#The_Mean_Square_Error_as_a_quality_criteria|$\text{Example 1}$]]: &nbsp; &nbsp;   $T_{\rm A}$&nbsp; is too large and&nbsp; $f_{\rm A}$&nbsp; is too small dimensioned.
+*Compared to the&nbsp; "Comparison System"&nbsp; according to&nbsp; [[Signal_Representation/Possible_Errors_when_using_DFT#The_mean_square_error_as_a_quality_criteria|$\text{Example 1}$]]: &nbsp; &nbsp;   $T_{\rm A}$&nbsp; is too large and&nbsp; $f_{\rm A}$&nbsp; is too small dimensioned.
 *The number of interpolation points is in both cases:&nbsp; $N = 16$.