Define signal y(t) = %∞

k=−∞ x(0.5t − 10k),

where
x(t) =
 e−2t t ≥ 1
0 t < 1

(a) Determine the constant a such that the
signal x(−2t +a) is borderline anticausal.
(b) Is the signal y(t) periodic? If so, determine
the period

We are given the signal \( y(t) \) defined as:

\[
y(t) = \sum_{k=-\infty}^{\infty} x(0.5t - 10k)
\]

where \( x(t) \) is a piecewise function:

\[
x(t) = \begin{cases}
e^{-2t} & \text{for } t \geq 1, \\
0 & \text{for } t < 1.
\end{cases}
\]

Let's address each part of the problem.

### Part (a): Determine the constant \( a \) such that \( x(-2t + a) \) is borderline anticausal.

#### Step 1: Analyze causality
- A signal is **causal** if it is nonzero only for \( t \geq 0 \).
- A signal is **anticausal** if it is nonzero only for \( t < 0 \).
- A signal is **borderline anticausal** if it transitions from being nonzero at \( t = 0 \) and becomes zero for all \( t > 0 \).

We need to shift and reflect \( x(t) \) in such a way that the resulting signal \( x(-2t + a) \) is nonzero for \( t \leq 0 \) and zero for \( t > 0 \).

#### Step 2: Set the argument of \( x \) to be equal to 1 for the borderline case
For \( x(-2t + a) \) to be nonzero at \( t = 0 \), the argument inside the function \( x(t) \) must be equal to 1 (since \( x(t) \) is nonzero only when \( t \geq 1 \)).

Thus, we set the argument equal to 1:

\[
-2(0) + a = 1 \implies a = 1.
\]

Now, let’s check the behavior for \( t = 0 \) and \( t > 0 \):
- When \( t = 0 \), \( x(-2(0) + 1) = x(1) = e^{-2} \), so the signal is nonzero.
- When \( t > 0 \), \( -2t + 1 < 1 \), so \( x(-2t + a) = 0 \) for all \( t > 0 \).

Thus, the constant \( a = 1 \) ensures that the signal is borderline anticausal.

### Part (b): Is the signal \( y(t) \) periodic? If so, determine the period.

#### Step 1: Analyze the structure of \( y(t) \)

The signal \( y(t) \) is a sum of shifted versions of \( x(0.5t - 10k) \). We need to determine if there is a fundamental period \( T \) such that \( y(t + T) = y(t) \) for all \( t \).

Each term \( x(0.5t - 10k) \) corresponds to a time shift by \( 10k \), which suggests a periodic structure. However, the time scaling factor of \( 0.5 \) in the argument modifies the period of the sum.

#### Step 2: Find the fundamental period
To determine the period, we note that the signal \( y(t) \) is composed of shifts by multiples of 10, but the argument is scaled by \( 0.5 \). Therefore, we need to solve for \( T \) such that:

\[
x(0.5(t + T) - 10k) = x(0.5t - 10k) \quad \forall k.
\]

This is satisfied if \( 0.5T = 10 \), leading to:

\[
T = 20.
\]

Thus, the signal \( y(t) \) is periodic with period \( T = 20 \).

---

### Summary:

- (a) The constant \( a = 1 \) makes the signal \( x(-2t + a) \) borderline anticausal.
- (b) The signal \( y(t) \) is periodic with period \( T = 20 \).

Would you like more details on any part? Here are some related questions for further exploration:

1. What is the difference between causal and anticausal signals?
2. How does scaling the time argument affect the period of a signal?
3. What are the properties of periodic signals in terms of Fourier series?
4. How can we analyze the frequency components of periodic signals?
5. How does shifting a signal by a constant affect its causality?

**Tip:** For signals with time scaling, always check how the scaling affects both the time shifts and the periodicity of the signal.

Define signal y(t) = ∑∞k=−∞ x(0.5t − 10k), where x(t) = e−2t t ≥ 1, 0 t < 1. (a) Determine the constant a such that the signal x(−2t + a) is borderline anticausal. (b) Is the signal y(t) periodic? If so, determine the period.

Explore how to determine the constant a for borderline anticausality of the signal x(-2t + a) and check the periodicity of the signal y(t). Learn key concepts in signal processing, including causality, time-shifting, and periodic functions. Suitable for undergraduate electrical engineering students.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation