Value of the integral \(\rm \int_{-\infty}^{\infty} \delta(t) \sin \left(\frac{t}{\sqrt{2}}\right) d t\) is:

Unit impulse function:

It is defined as, \(\delta \left( t \right) = \left\{ {\begin{array}{*{20}{c}} {\infty ,\;\;t = 0}\\ {0,\;\;t \ne 0} \end{array}} \right.\)

The discrete-time version of the unit impulse is defined by

\(\delta \left[ n \right] = \left\{ {\begin{array}{*{20}{c}} {1,\;\;n = 0}\\ {0,\;\;n \ne 0} \end{array}} \right.\)

Properties:

1. \(\mathop \smallint \limits_{ - \infty }^\infty \delta \left( t \right)dt = 1\)

2. \(\delta \left( {at} \right) = \frac{1}{{\left| a \right|}}\delta \left( t \right)\)

3. x(t) δ(t – t0) = x(t0)

4. \(\mathop \smallint \limits_{ - \infty }^\infty x\left( t \right)\delta \left( {t - {t_o}} \right)dt = x\left( {{t_0}} \right)\)

5. \(\mathop \smallint \limits_{ - \infty }^\infty f\left( t \right)\delta \left( {at + b} \right)dt = \mathop \smallint \limits_{ - \infty }^\infty f\left( t \right)\frac{1}{{\left| a \right|}}\delta \left( {t + \frac{b}{a}} \right)dt\)

6. \(\mathop \smallint \limits_{ - \infty }^\infty x\left( t \right){\delta ^n}\left( {t - {t_o}} \right)dt = {\left. {\frac{{{d^n}x}}{{d{t^n}}}} \right|_{t = {t_0}}}\)

Calculation:

Apply the sifting property of the delta function:

\( \int_{-\infty}^{\infty} \delta(t) \cdot \sin\left( \frac{t}{\sqrt{2}} \right) dt = \sin\left( \frac{0}{\sqrt{2}} \right) = \sin(0) = 0 \)