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ISRO Scientist EC 2015 Official Paper

Option 3 : x(n – 1)

CT 1: Current Affairs (Government Policies and Schemes)

54676

10 Questions
10 Marks
10 Mins

__Concept:__

Shifting property of convolution

Convolution of the signal x(t) with the impulse gives the same signal as a result after the convolution.

x(t) ∗ δ(t) = x(t)

x(t) ∗ δ(t – t0) = x(t – t0)

Substitute ‘t – t0’ in the place of ‘t’

__calculation:__

Given:

x(n) = {1, 1, 1, 1, 0.5, 0.5}

y(n) = conv (δ(n – 1), x(n))

Using property of impilse function:

y(n) = x(n - 1)

__Properties of δ(t):__

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

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

3. x(t) δ(t – t0) = x(t0)

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

5. \(\mathop \smallint \limits_{ - \infty }^\infty f\left( t \right)δ \left( {at + b} \right)dt\)

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

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

__Properties of δ(n):__

1. x[n] δ[n] = x[0] δ[n]

2. \(\mathop \sum \limits_{n = - \infty }^{n = \infty } x\left[ n \right]δ \left[ n \right]\)

\( = \mathop \sum \limits_{n = - \infty }^{n = \infty } x\left[ 0 \right]δ \left[ n \right] = x\left[ 0 \right]\)

3. x[n] δ[n – n0] = x[n0] δ[n – n0]

4. \(\mathop \sum \limits_{n = - \infty }^{n = \infty } x\left[ n \right]δ \left[ {n - {n_0}} \right]\)

\( = \mathop \sum \limits_{n = - \infty }^{n = \infty } x\left[ {{n_0}} \right]δ \left[ {n - {n_0}} \right] = x\left[ {{n_0}} \right]\)

5. δ[an] = δ[n]