Signals & System Lecture 12

Notes from 5-9-2024

1. Time Shifting Properties

• If $x(t)\stackrel{LT}{\to }X(s)$, ROC:
  • Delay: $x(t-K)\stackrel{LT}{\to }{e}^{-Ks}X(s)$, same ROC
  • Advance: $x(t+K)\stackrel{LT}{\to }{e}^{+Ks}X(s)$, same ROC

2. Multiplication

• If $x(t)\stackrel{LT}{\to }X(s)$, ROC:

  • $t\cdot x(t)\stackrel{LT}{\to }-\frac{d}{ds}X(s)$, same ROC

  Always draw ROC whether mentioned or not.

3. Multiplication by Exponential Property

• If $x(t)\stackrel{LT}{\to }X(s)$, ROC:
  • $e^{at}\cdot x(t)\stackrel{LT}{\to }X(s-a)$, ROC may change
  • $e^{-at}\cdot x(t)\stackrel{LT}{\to }X(s+a)$, new ROC

4. Scaling in Time Domain

• If $x(at)\stackrel{LT}{\to }\frac{1}{|a|}X\left(\frac{s}{a}\right)$, ROC:
  • For $a>1$, ROC is scaled by a factor of $a$
  • For $a<1$, ROC is compressed

5. Time Differentiation

• If $\frac{d}{dt}x(t)\stackrel{LT}{\to }sX(s)$, ROC:
  • The ROC remains unchanged, provided the system is causal.

6. Time Integration

• If ${\int }_{-\infty }^{t}x(\tau )d\tau \stackrel{LT}{\to }\frac{X(s)}{s}$, ROC:
  • ROC might change based on the behavior of the integrand.

References

Information

• date: 2024.09.09
• time: 10:35