Consider a continuous-time signal
$x(t) = – t^2 \{u(t + 4) – u(t – 4)\}$
where $u(t)$ is the continuous-time unit step function. Let $\delta(t)$ be the continuous-time unit impulse function. The value of
$\int_{-\infty}^{\infty} x(t) \delta(t + 3)dt$ is
Evaluating the Integral using Sifting Property
The problem requires finding the value of the integral $ \int_{-\infty}^{\infty} x(t) \delta(t + 3)dt $, where $ x(t) = – t^2 \{u(t + 4) – u(t – 4)\} $ and $ \delta(t) $ is the Dirac delta function.
Understanding the Signal x(t)
The term $ \{u(t + 4) – u(t – 4)\} $ defines a rectangular pulse that equals 1 for $ -4 < t < 4 $ and is 0 otherwise. Therefore, the signal $ x(t) $ can be expressed as:
- $ x(t) = – t^2 $ for $ -4 < t < 4 $
- $ x(t) = 0 $ otherwise
Applying the Sifting Property
The sifting property of the Dirac delta function states that $ \int_{-\infty}^{\infty} f(t) \delta(t - a)dt = f(a) $. This property allows us to evaluate the integral by substituting the value 'a' into the function $ f(t) $, provided 'a' is within the function's non-zero region.
In this integral, $ f(t) = x(t) $ and the impulse is shifted to $ a = -3 $, as $ \delta(t + 3) $ is equivalent to $ \delta(t - (-3)) $.
Applying the sifting property, the integral becomes:
$ \int_{-\infty}^{\infty} x(t) \delta(t + 3)dt = x(-3) $Calculating the Value
We need to find the value of $ x(t) $ at $ t = -3 $. Since $ -4 < -3 < 4 $, the function $ x(t) $ is defined as $ – t^2 $ at this point.
- Substitute $ t = -3 $ into $ x(t) = – t^2 $: $ x(-3) = -(-3)^2 $
- Calculate the result: $ x(-3) = -(9) = -9 $
Therefore, the value of the integral is $ -9 $.
Important Questions from Standard Signals
Inverse Fourier Transform of δ(ω - ω 0) is ______.
The following statements relate to sampling distributions. Choose the correct code for the statements being correct or incorrect.
Statement I: Sampling distribution of mean is normally distributed irrespective of the type of population distribution and size of samples.
Statement II : The standard deviation of the sampling distribution of mean is less than the standard deviation of the population distribution.
The value of \(\mathop \smallint \limits_{ - \infty }^{ + \infty } {e^{ - t}}\delta \left( {2t - 2} \right)dt\), where \(\delta \left( t \right)\) is the Dirac delta function, is
- \(\mathop \smallint \nolimits_{ - 7}^2 \left( {{t^2} + {t^3} + 1} \right)\delta \left( {t - 3} \right)dt = \_\_\_\_\)
Which of the following points CANNOT be observed about a unit impulse function if it is assumed in the form of a pulse?
