Q4Signals and Systems

Question

A signal x[n] with Fourier transform $x (e^{j ω})$ has the property that: $(x [n] k = - \infty \sum \infty δ [n - 3 k]) * (\frac{sin \frac{π}{3} n}{\frac{π}{3} n}) = x [n]$ For what values of w is it guaranteed that $x (e^{j ω}) = 0$ .

Answer 1

The given equation represents the process of discrete-time sampling and interpolation (reconstruction):

1. The term $x [n] \sum δ [n - 3 k]$ represents sampling the discrete-time signal $x [n]$ every $N = 3$ samples. This process creates periodic replicas of the original frequency spectrum $X (e^{j ω})$ shifted by integer multiples of $2 π /3$ .

2. The sampled signal is then convolved with an ideal discrete-time low-pass filter (represented by the sinc function) having a cutoff frequency of $ω_{c} = π /3$ .

3. The equation states that the output of this low-pass filter exactly equals the original sequence $x [n]$ .

According to the Nyquist Sampling Theorem, for perfect reconstruction from sampled data without aliasing distortion, the original signal's frequency spectrum must not overlap when replicated. Therefore, the signal must be strictly band-limited to half the sampling frequency.

Since the sampling period here is $N = 3$ , the sampling frequency is $2 π /3$ . The Nyquist limit is half of this, which is $π /3$ . To avoid aliasing and ensure perfect reconstruction via the ideal low-pass filter, the original spectrum $X (e^{j ω})$ must contain no frequency components above $π /3$ .

Thus, it is guaranteed that $X (e^{j ω}) = 0$ for $\frac{π}{3} \leq ∣ ω ∣ \leq π$ .

Q4Signals and Systems

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