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Continuous-Time Causality 3. Determine whether or not the systems with these fr...
Apr 25, 2024
Continuous-Time Causality 3. Determine whether or not the systems with these frequency responses are causal. (a) H(f)=sinc⁡(f)\mathrm{H}(f)=\operatorname{sinc}(f) (b) H(f)=sinc⁡(f)e−jπf\mathrm{H}(f)=\operatorname{sinc}(f) e^{-j \pi f} (c) H(jω)=rect⁡(ω)\mathrm{H}(j \omega)=\operatorname{rect}(\omega) (d) H(jω)=rect⁡(ω)e−jω\mathrm{H}(j \omega)=\operatorname{rect}(\omega) e^{-j \omega} (e) H(f)=A\mathrm{H}(f)=A (f) H(f)=Aej2πf\mathrm{H}(f)=A e^{j 2 \pi f} (g) H(jω)=1−e−jω\mathrm{H}(j \omega)=1-e^{-j \omega} (h) H(f)=rect⁡(f/20)e−j40πf\mathrm{H}(f)=\operatorname{rect}(f / 20) e^{-j 40 \pi f} (i) H(ejΩ)=ejΩejΩ−0.9\mathrm{H}\left(e^{j \Omega}\right)=\frac{e^{j \Omega}}{e^{j \Omega}-0.9} (j) H(ejΩ)=ej2ΩejΩ−1.3\mathrm{H}\left(e^{j \Omega}\right)=\frac{e^{j 2 \Omega}}{e^{j \Omega}-1.3}
B
Key Concept
Causality and the Fourier Transform
Explanation
A system is causal if its impulse response is zero for all negative times. In the frequency domain, a system is causal if its frequency response is analytic and has no poles or singularities in the upper half of the complex frequency plane. The given frequency response H(f)=sinc⁡(f)e−jπfH(f)=\operatorname{sinc}(f) e^{-j \pi f} corresponds to a time-shifted sinc function in the time domain, which is causal because the time shift corresponds to a linear phase term in the frequency domain that does not introduce any non-causality.
F
Key Concept
Non-causality and Exponential Functions
Explanation
The frequency response H(f)=Aej2πfH(f)=A e^{j 2 \pi f} represents a non-causal system because the exponential term ej2πfe^{j 2 \pi f} corresponds to a time advance in the time domain. A time advance implies that the system would respond before the input is applied, which is non-causal.
G
Key Concept
Causality and Poles in the Frequency Response
Explanation
The frequency response H(jω)=1−e−jωH(j \omega)=1-e^{-j \omega} is causal because it can be associated with an impulse response that is zero for negative times. The term e−jωe^{-j \omega} corresponds to a delay in the time domain, which is a property of causal systems.
H
Key Concept
Causality and Phase Terms in Frequency Response
Explanation
The frequency response H(f)=rect⁡(f/20)e−j40πfH(f)=\operatorname{rect}(f / 20) e^{-j 40 \pi f} is causal. The rectangular function rect⁡(f/20)\operatorname{rect}(f / 20) limits the bandwidth, and the exponential term e−j40πfe^{-j 40 \pi f} represents a linear phase shift, which corresponds to a time delay in the time domain, preserving causality.
I
Key Concept
Causality and Poles in the Z-Transform
Explanation
The system described by H(ejΩ)=ejΩejΩ−0.9H\left(e^{j \Omega}\right)=\frac{e^{j \Omega}}{e^{j \Omega}-0.9} is causal because the pole at 0.90.9 is inside the unit circle in the z-plane. For discrete-time systems, causality is ensured if all poles of the z-transform are inside the unit circle.
J
Key Concept
Non-causality and Poles in the Z-Transform
Explanation
The system described by H(ejΩ)=ej2ΩejΩ−1.3H\left(e^{j \Omega}\right)=\frac{e^{j 2 \Omega}}{e^{j \Omega}-1.3} is non-causal because the pole at 1.31.3 is outside the unit circle in the z-plane. For discrete-time systems, a pole outside the unit circle implies that the system is non-causal, as it would require future input values to compute the current output.
Continuous-Time Causality 3. Determine whether or not the systems with these frequency responses are causal. (a) H(f)=sinc⁡(f)\mathrm{H}(f)=\operatorname{sinc}(f) (b) H(f)=sinc⁡(f)e−jπf\mathrm{H}(f)=\operatorname{sinc}(f) e^{-j \pi f} (c) H(jω)=rect⁡(ω)\mathrm{H}(j \omega)=\operatorname{rect}(\omega) (d) H(jω)=rect⁡(ω)e−jω\mathrm{H}(j \omega)=\operatorname{rect}(\omega) e^{-j \omega} (e) H(f)=A\mathrm{H}(f)=A (f) H(f)=Aej2πf\mathrm{H}(f)=A e^{j 2 \pi f} (g) H(jω)=1−e−jω\mathrm{H}(j \omega)=1-e^{-j \omega} (h) H(f)=rect⁡(f/20)e−j40πf\mathrm{H}(f)=\operatorname{rect}(f / 20) e^{-j 40 \pi f} (i) H(ejΩ)=ejΩejΩ−0.9\mathrm{H}\left(e^{j \Omega}\right)=\frac{e^{j \Omega}}{e^{j \Omega}-0.9} (j) H(ejΩ)=ej2ΩejΩ−1.3\mathrm{H}\left(e^{j \Omega}\right)=\frac{e^{j 2 \Omega}}{e^{j \Omega}-1.3}
A
Key Concept
Causality and Frequency Response
Explanation
A system is causal if its impulse response is zero for all negative times. The sinc function in the frequency domain corresponds to a time-domain function that is non-zero for negative times, thus the system is non-causal.
B
Key Concept
Causality and Time Shifting
Explanation
Multiplying by e−jπfe^{-j \pi f} corresponds to a time shift in the time domain. Since the sinc function is non-causal, shifting it in time does not make it causal.
C
Key Concept
Rectangular Function and Causality
Explanation
The rectangular function, or rect function, in the frequency domain corresponds to a sinc function in the time domain, which is non-zero for negative times, indicating a non-causal system.
D
Key Concept
Time Shifting in Frequency Domain
Explanation
Multiplying by e−jωe^{-j \omega} corresponds to a time shift. Since the rect function is non-causal in the time domain, shifting it does not result in a causal system.
E
Key Concept
Constant Frequency Response and Causality
Explanation
A constant frequency response corresponds to a delta function in the time domain, which is causal since it is zero for all negative times.
F
Key Concept
Exponential in Frequency Domain and Non-causality
Explanation
An exponential term like ej2πfe^{j 2 \pi f} corresponds to a time shift, which does not affect causality. Since a constant AA is causal, the system remains causal after the shift.
G
Key Concept
Frequency Response and Impulse Response
Explanation
The term 1−e−jω1-e^{-j \omega} corresponds to a time-domain function that is non-zero for negative times, indicating a non-causal system.
H
Key Concept
Time Shifting and Frequency Scaling
Explanation
The term e−j40πfe^{-j 40 \pi f} represents a time shift, and the rect function scaled by 1/201/20 is non-causal. The combination remains non-causal.
I
Key Concept
Poles and Z-Transform Causality
Explanation
For a system to be causal, all poles of its Z-transform must lie inside the unit circle. The given H has a pole at 0.90.9, which is inside the unit circle, so the system is causal.
J
Key Concept
Poles and Z-Transform Causality
Explanation
If a pole of the Z-transform lies outside the unit circle, the system is non-causal. The given H has a pole at 1.31.3, which is outside the unit circle, indicating a non-causal system.
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