Signals and Systems

Signals

ELEMENTARY SIGNALS

Properties of Unit Impulse function:

🌾Delta function is an even function:

A function $f(t)$ is said to be an even function, if it remains unchanged after a time reversal $\big(f(-t)\big)$ $$f(t)=f(-t)$$

To understand this property, we may use rectangular pulse version of delta function $\delta(t)$, 

By seeing the above figure, we can say, it is symmetric about amplitude axis $(y-axis)$.

 

Time reversal can be done by replacing $t$ by $-t$ (i.e.) by flipping the above function about the amplitude axis $$\delta(t) \xrightarrow{\text{time reversal}} \delta(-t)$$.

Flipping the above figure about $y-axis$ results in same.

By comparing the above two figures, we can say$$\delta(t)=\delta(-t)$$ 

Since

  • the delta function is symmetric about amplitude axis $(y-axis)$
  • time reversal of the function results in same function $\delta(t)=\delta(-t)$

We can conclude that delta function is an even function.

🌾Scaling property:

To understand this property, we may use rectangular pulse version of delta function $\delta(t)$, 

We can time-scale the above function by replacing $t$ by $at$, $$\delta(t) \xrightarrow{\text{time scaling}} \delta(at)$$

From the above figure, we can say the width of the pulse is $T=\epsilon$, since $t$ becomes $’at’$ in time scaling, the width of the pulse will be $$aT=\epsilon$$ $$T=\frac{\epsilon}{a}$$

Case(i): When $a\lt 1$

Let us assume $a=0.5$, 

The width of the pulse will be$$T=\frac{\epsilon}{a}$$ $$T=\frac{\epsilon}{0.5}$$ $$T=2\epsilon$$

The width of the pulse gets increased, hence it is called time expansion

Case(ii): When $a\gt 1$

 Let us assume $a=2$, 

The width of the pulse will be$$T=\frac{\epsilon}{a}$$ $$T=\frac{\epsilon}{2}$$

The width of the pulse gets reduced, hence it is called time compression.

Remember, area under the integral of delta function is always equal to one.
 
From the above two cases, we can say when we attempt to compress the delta function in time-scale, it expands in amplitude-scale or vice-versa. Hence we can safely say, time-scaling of delta function will automatically leads to amplitude scaling.

Time scaling of delta function $\to$ Amplitude scaling

Mathematically, $$\delta(at)=\frac{1}{|a|}\delta(t)$$
Proof:
As we know, the area under the integral is always one for delta function $$\int_{-\infty}^{\infty}\delta(at)dt=1 \tag{1}$$
Let $at=\tau$, then $d\tau=a\;dt$, $dt=\frac{d\tau}{a}$ $$\int_{-\infty}^{\infty}\delta(\tau)\frac{d\tau}{a}=1$$ Rearranging, $$\int_{-\infty}^{\infty}\frac{\delta(\tau)}{a}{d\tau}=1\tag{2}$$ 
By comparing equations $(1)$ and $(2)$, we can say $$\delta(at)=\frac{\delta(t)}{a}$$
Since, delta function is an even function $\delta(-at)$ results in same $$\delta(-at)\to \delta(at)=\frac{\delta(t)}{a}$$ Hence, we can rewrite the above equation as, $$\delta(at)=\frac{\delta(t)}{|a|}$$

Time scaling of delta function will automatically leads to Amplitude scaling, in order to maintain the fact that area under the integral of delta function is always equal to one $$\delta(at)=\frac{\delta(t)}{|a|}$$

Practice Question

🌾Prove delta function is an even function using scaling property.

For delta function to be an even function, we have to prove $$\delta(-t)=\delta(t)$$

Proof:

From scaling proprety of delta function, $$\delta(at)=\frac{1}{|a|}\delta(t)$$ Substituting $a=-1$ gives us $$\delta(-t)=\frac{1}{|-1|}\delta(t)$$ $$\delta(-t)=\frac{1}{|1|}\delta(t)$$ $$\delta(-t)=\delta(t)$$