View source for Aufgaben:Exercise 1.5Z: Sinc-shaped Impulse Response


{{quiz-Header|Buchseite=Linear_and_Time_Invariant_Systems/Some_Low-Pass_Functions_in_Systems_Theory}}

[[File:P_ID857__LZI_Z_1_5.png|right|frame|$\rm sinc$–shaped impulse response]]
The impulse response of a linear time-invariant (and non-causal) system was determined as follows (see graph):
:$$h(t) = 500\hspace{0.1cm}{ {\rm s}}^{-1}\cdot{\rm sinc}\big[{t}/({ 1\hspace{0.1cm}{\rm ms}})\big] .$$
The output signals&nbsp; $y(t)$ should be computed if various cosine oscillations of different frequency&nbsp; $f_0$&nbsp; are applied to the input:
:$$x(t) = 4\hspace{0.05cm}{\rm V}\cdot {\rm cos}(2\pi \cdot  f_0
\cdot t ) .$$









''Please note:'' 
*The exercise belongs to the chapter&nbsp; [[Linear_and_Time_Invariant_Systems/Some_Low-Pass_Functions_in_Systems_Theory|Some Low-Pass Functions in Systems Theory]].
*The solution can be found in the time domain or in the frequency domain. In the sample solution you will find both approaches. 
*The following definite integral is given:
:$$\int_{ 0 }^{ \infty } \frac{\sin(u) \cdot \cos(a \cdot u)}{u}   \hspace{0.15cm}{\rm
 d}u = \left\{ \begin{array}{c} \pi/2 \\  \pi/4 \\ 0 \\  \end{array} \right.\quad \quad
\begin{array}{c}   {\rm{f\ddot{u}r}}  \\ {\rm{f\ddot{u}r}}
\\   {\rm{f\ddot{u}r}}  \\ \end{array}\begin{array}{*{20}c}{ |a|  < 1,}  \\{ |a|  = 1,}  \\
{ |a|  > 1.}  \\ \end{array}$$
 



===Questions===

<quiz display=simple>
{Compute the frequency response&nbsp; $H(f)$&nbsp; of the LTI system. What is the equivalent bandwidth and the direct signal (DC) transmission factor?
|type="{}"}
$\Delta f \ =\ $ { 1 3% } $\ \rm kHz$
$H(f = 0) \ =\ $ { 0.5 3% }


{What is the signal value of the output signal&nbsp; $y(t)$&nbsp; at time&nbsp; $t = 0$&nbsp; if the input is cosine-shaped and of frequency&nbsp; $\underline{f_0 = 1\ \rm  kHz}$?
|type="{}"}
$y(t = 0) \  = \ $ { 0. } $\ \rm V$


{What is the signal value of the output signal&nbsp; $y(t)$&nbsp; at time&nbsp; $t = 0$&nbsp; if the input is cosine-shaped and of frequency&nbsp; $\underline{f_0 = 0.1\ \rm  kHz}$?
|type="{}"}
$y(t = 0) \  =\ $ { 2 3% } $\ \rm V$


{What is the signal value of the output signal&nbsp; $y(t)$&nbsp; at timee&nbsp; $t = 0$&nbsp; if the input is cosine-shaped and of frequency&nbsp; $\underline{f_0 = 0.5\ \rm  kHz}$?
|type="{}"}
$y(t = 0) \  = \ $ { 1 3% } $\ \rm V$

</quiz>

===Solution===
{{ML-Kopf}}
'''(1)'''&nbsp; A comparison with the equations on the page&nbsp; [[Linear_and_Time_Invariant_Systems/Some_Low-Pass_Functions_in_Systems_Theory#Ideal_low-pass_filter_–_Rectangular-in-frequency|Ideal low-pass filter]] or applying the&nbsp; [[Signal_Representation/Fourier_Transform_and_Its_Inverse#The_second_Fourier_integral |inverse Fourier transformation]]&nbsp; shows that&nbsp; $H(f)$&nbsp; is an ideal low-pass filter:
:$$H(f) = \left\{ \begin{array}{c} \hspace{0.25cm}K  \\  K/2 \\ 0 \\  \end{array} \right.\quad \quad \begin{array}{*{10}c}   {\rm{f\ddot{u}r}}  \\ {\rm{f\ddot{u}r}}
\\   {\rm{f\ddot{u}r}}  \\ \end{array}\begin{array}{*{20}c}
{\left| \hspace{0.005cm} f\hspace{0.05cm} \right| < \Delta f/2,}  \\
{\left| \hspace{0.005cm}f\hspace{0.05cm} \right| = \Delta f/2,}  \\
{\left|\hspace{0.005cm} f \hspace{0.05cm} \right| > \Delta f/2.}  \\
\end{array}$$
*The equidistant zero-crossings of the impulse response occur at an interval of&nbsp; $Δt = 1 \ \rm ms$&nbsp;. 
*From this it follows that the equivalent bandwidth is&nbsp; $Δf \rm \underline{ = 1 \ \rm kHz}$.&nbsp; 
*If&nbsp; $K = 1$ was true, then&nbsp; $h(0) = Δf = 1000 \cdot \rm 1/s$&nbsp; should hold. 
*Because of the given&nbsp; $h(0) = 500 \cdot{\rm 1/s} = Δf/2$&nbsp; the direct signal (DC) transmission factor thus is&nbsp; $K = H(f = 0) \; \rm \underline{= 0.5}$.



'''(2)'''&nbsp; This problem is most easily solved in the spectral domain. 
*For the output spectrum the following holds: &nbsp; $Y(f) =  X(f)\cdot H(f) .$
*$X(f)$&nbsp; consists of two Dirac functions at&nbsp; $± f_0$ each with weight&nbsp; $A_x/2 =2 \hspace{0.08cm}\rm V$. 
*For&nbsp; $f = f_0 = 1 \ {\rm kHz} > Δf/2$&nbsp;, however&nbsp; $H(f) = 0$ holds, such that&nbsp; $Y(f) = 0$&nbsp; and hence also &nbsp; $y(t) = 0$&nbsp; &nbsp;  ⇒  &nbsp;  $\underline{y(t = 0) = 0}$.


The solution in the time domain is based on convolution:
:$$y(t) = x (t) * h (t) = \int_{ - \infty }^{ + \infty } {h ( \tau  )}  \cdot
 x ( {t - \tau } ) \hspace{0.1cm}{\rm d}\tau.$$
*At time&nbsp; $t = 0$&nbsp; the following is obtained considering the symmetry of the cosine function:
:$$y(t = 0 ) = \frac{A_x \cdot \Delta f}{2} \cdot \int_{ - \infty }^{ + \infty } {\rm si} ( \pi \cdot \Delta f \cdot \tau  )  \cdot
 {\rm cos}(2\pi \cdot  f_0
\cdot \tau ) \hspace{0.1cm}{\rm d}\tau.$$
*With the substitution&nbsp; $u = π · Δf · τ$&nbsp;, this can also be formulated as follows:
:$$y(t = 0 ) = \frac{A_x }{\pi} \cdot \int_{ 0 }^{ \infty } \frac{\sin(u) \cdot \cos(a \cdot u)}{u}   \hspace{0.15cm}{\rm d}u .$$
*Here, the constant is&nbsp; $a = 2f_0/Δf = 2$. With this value, the given integral yields zero: &nbsp; $y(t = 0 ) = {A_y } = 0.$



'''(3)'''&nbsp; The frequency response has the value&nbsp; $K = 0.5$ at&nbsp; $f = f_0 = 100 \ \rm Hz$&nbsp; according to the calculations for subtask&nbsp; '''(1)'''&nbsp;. Therefore, 
:$$A_y = A_x/2 = 2\ \rm  V$$ is obtained.
*The same result is obtained by convolution according to the above equation. 
*For&nbsp; $a = 2f_0/Δf = 0.2$&nbsp; the integral is equal to&nbsp; $π/2$&nbsp; and one obtains
:$$y(t = 0 ) = {A_y } = \frac{A_x}{\pi} \cdot \frac{\pi}{2} = \frac{A_x}{2} \hspace{0.15cm}\underline{= 2\,{\rm V}}.$$


'''(4)'''&nbsp; The transition from the band-pass to the band-stop is exactly at&nbsp; $f = 0.5   \ \rm kHz$&nbsp; and for this singular location the following holds: 
:$$H(f = f_0) = K/2.$$ 
*Thus, the amplitude of the output signal is only half as large as calculated in subtask&nbsp; '''(3)'''&nbsp;, namely&nbsp; $A_y \;  \underline{= 1  \, \rm V}$. 
*The same result is obtained with&nbsp;$a = 2f_0/Δf = 1$&nbsp; by convolution.
{{ML-Fuß}}



[[Category:Linear and Time-Invariant Systems: Exercises|^1.3 Some Low-Pass Functions in Systems Theory^]]