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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)

\mathrm{H}(f)=\operatorname{sinc}(f)

(b)

\mathrm{H}(f)=\operatorname{sinc}(f) e^{-j \pi f}

(c)

\mathrm{H}(j \omega)=\operatorname{rect}(\omega)

(d)

\mathrm{H}(j \omega)=\operatorname{rect}(\omega) e^{-j \omega}

(e)

\mathrm{H}(f)=A

(f)

\mathrm{H}(f)=A e^{j 2 \pi f}

(g)

\mathrm{H}(j \omega)=1-e^{-j \omega}

(h)

\mathrm{H}(f)=\operatorname{rect}(f / 20) e^{-j 40 \pi f}

(i)

\mathrm{H}\left(e^{j \Omega}\right)=\frac{e^{j \Omega}}{e^{j \Omega}-0.9}

(j)

\mathrm{H}\left(e^{j \Omega}\right)=\frac{e^{j 2 \Omega}}{e^{j \Omega}-1.3}

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)=\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.

Key Concept

Non-causality and Exponential Functions

Explanation

The frequency response

H(f)=A e^{j 2 \pi f}

represents a non-causal system because the exponential term

e^{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.

Key Concept

Causality and Poles in the Frequency Response

Explanation

The frequency response

H(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^{-j \omega}

corresponds to a delay in the time domain, which is a property of causal systems.

Key Concept

Causality and Phase Terms in Frequency Response

Explanation

The frequency response

H(f)=\operatorname{rect}(f / 20) e^{-j 40 \pi f}

is causal. The rectangular function

\operatorname{rect}(f / 20)

limits the bandwidth, and the exponential term

e^{-j 40 \pi f}

represents a linear phase shift, which corresponds to a time delay in the time domain, preserving causality.

Key Concept

Causality and Poles in the Z-Transform

Explanation

The system described by

H\left(e^{j \Omega}\right)=\frac{e^{j \Omega}}{e^{j \Omega}-0.9}

is causal because the pole at

0.9

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.

Key Concept

Non-causality and Poles in the Z-Transform

Explanation

The system described by

H\left(e^{j \Omega}\right)=\frac{e^{j 2 \Omega}}{e^{j \Omega}-1.3}

is non-causal because the pole at

1.3

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.