Two periodic signals x(𝑡) and 𝑦(𝑡) have the same fundamental period of 3 seconds. Consider the signal 𝑧(𝑡) = 𝑥(−𝑡) + 𝑦(2𝑡 + 1). The fundamental period of 𝑧(𝑡) in seconds is

Understanding Periodic Signals and Their Periods

A signal is considered periodic if it repeats its pattern over a specific time interval. This minimum time interval is called the fundamental period, often denoted by $T$. For a periodic signal $x(t)$ with fundamental period $T$, the property $x(t)=x(t+T)$ holds true for all values of $t$. The question involves two such signals, $x(t)$ and $y(t)$, both having a fundamental period of 3 seconds.

Analyzing Signal Transformations for Periodicity

The signal $z(t)$ is formed by transforming $x(t)$ and $y(t)$. We need to understand how these transformations affect the fundamental period:

- **Time Reversal (x(t)):** If a signal $x(t)$ has a fundamental period $T$, the signal x(t) also has the same fundamental period $T$. The reversal operation does not alter the time duration of the repeating pattern.
- **Time Scaling ($y(at)$):** If a signal $y(t)$ has a fundamental period $T$, the signal $y(at)$ has a new fundamental period of $\frac{T}{|a|}$. This is because the signal is compressed or stretched along the time axis by a factor of $a$.
- **Time Shifting ($y(t+b)$):** Shifting a signal in time by adding a constant $b$, like in $y(t+b)$, does not change its fundamental period. The pattern repeats at the same rate, just starting at a different time point.
- **Sum of Periodic Signals ($x_1(t)+x_2(t)$):** When two periodic signals, $x_1(t)$ with period $T_1$ and $x_2(t)$ with period $T_2$, are added, the resulting signal's fundamental period is the Least Common Multiple (LCM) of their individual periods, i.e., $T_z=\text{LCM}(T_1,T_2)$. This assumes the sum is indeed periodic.

Calculating the Fundamental Period of z(t)

We are given z(t) = x(t) + y(2t + 1). Let's find the periods of the individual components:

**Period of the first component: x(t) **

The original signal $x(t)$ has a fundamental period $T_x=3$ seconds.

Applying the time reversal transformation, the period of x(t) remains unchanged.

Let T_x(t) denote the fundamental period of x(t).

T_x(t) = T_x = 3 seconds

**Period of the second component: $y(2t+1)$ **

The original signal $y(t)$ has a fundamental period $T_y=3$ seconds.

The transformation is $y(2t+1)$. Here, $a=2$ (time scaling) and $b=1$ (time shifting).

Using the time scaling rule, the period of $y(at+b)$ is $\frac{T}{|a|}$.

Let $T_{y(2t+1)}$ denote the fundamental period of $y(2t+1)$.

$$T_{y(2t+1)}=\frac{T_y}{|a|}=\frac{3}{|2|}=\frac{3}{2}=1.5\text{ seconds}$$

**Finding the Period of the Sum $z(t)$ **

The signal $z(t)$ is the sum of x(t) (period $T_1=3$ seconds) and $y(2t+1)$ (period $T_2=1.5$ seconds).

The fundamental period of $z(t)$, denoted $T_z$, is the LCM of $T_1$ and $T_2$.

$$T_z=\text{LCM}(T_1,T_2)=\text{LCM}(3,1.5)$$

Calculating the LCM

To calculate the LCM of 3 and 1.5, we can express 1.5 as a fraction:

$1.5=\frac{3}{2}$

So we need to find $\text{LCM}(3,\frac{3}{2})$.

The general formula for the LCM of two fractions $\frac{p}{q}$ and $\frac{r}{s}$ is $\frac{\text{LCM}(p,r)}{\text{GCD}(q,s)}$.

In our case, $3$ can be written as $\frac{3}{1}$. So, $p=3,q=1,r=3,s=2$.

$$T_z=\frac{\text{LCM}(3,3)}{\text{GCD}(1,2)}$$

Calculating the numerator and denominator:

- $\text{LCM}(3,3)=3$
- $\text{GCD}(1,2)=1$

Substituting these values back:

$$T_z=\frac{3}{1}=3\text{ seconds}$$

Alternatively, listing the multiples:

- Multiples of 3 seconds: 3, 6, 9, ...
- Multiples of 1.5 seconds: 1.5, 3.0, 4.5, 6.0, ...

The smallest value that appears in both lists is 3.

Conclusion

The fundamental period of the signal z(t) = x(t) + y(2t + 1) is determined to be 3 seconds.