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GATE 2019 EC – Question 39

Networks, Signals and Systems · Discrete-time Signals · 2 marks · Multiple choice

It is desired to find a three-tap causal filter which gives zero signal as an output to an input of the form

$$x[n]=c_1\exp\left(-\frac{j\pi n}{2}\right)+c_2\exp\left(\frac{j\pi n}{2}\right),$$

where $c_1$ and $c_2$ are arbitrary real numbers. The desired three-tap filter is given by $h[0]=1$, $h[1]=a$, $h[2]=b$ and $h[n]=0$ for $n<0$ or $n>2$.

What are the values of the filter taps $a$ and $b$ if the output is $y[n]=0$ for all $n$, when $x[n]$ is as given above?

  1. $a=1,b=1$
  2. $a=0,b=-1$
  3. $a=-1,b=1$
  4. $a=0,b=1$

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Show answer and explanation

Correct answer: (D) $a=0,b=1$

Explanation

The filter must have zero gain at $\omega=\pm\frac\pi2$: $H(e^{j\pi/2})=1+ae^{-j\pi/2}+be^{-j\pi}=1-ja-b=0$. Real and imaginary parts give $a=0$ and $b=1$.