Comparison between ideal, linear and lowpass filter interpolators

[theos]

Homework Statement

Q3. Sampling and reconstruction
A) It is common practice to use low order low pass filters instead of a Sinc filter, at the expense
of a higher sampling rate. Here we investigate how to do this.
Compare:
a) the ideal reconstruction of a low pass sampled signal, which can be achieved with an ideal
interpolator with V(t)=Σ(of n) (Vn sinc(πFs(t-tn ))
, where tn are the sampling instants and vn the voltages measured at these instants
with
b) the reconstruction that can be achieved with a linear interpolation
V(t) = Vn−1*(tn-t)/T+Vn*(t-tn-1)/T
between two samples taken at times tn−1 and tn .
Derive a condition on the sampling rate so that the difference between the two interpolations is smaller that a specified error δ .

Generalise this result to specify the required sampling rate and pole position of a 1st order
low pass filter used for the interpolation.

HINT: This is probably much easier to do in the frequency domain, by comparing the power
of the reconstructed signal to the original. After all, the reconstructed signal is the
convolution of the samples with the interpolating function.

What is the Fourier transform of a linear interpolator? How does it differ from the Fourier
transform of the Sinc interpolator (which is a square box in the frequency domain)?

What is the Fourier transform of a first order Low Pas filter, and how does it differ from that
of the ideal interpolator?

Homework Equations

The Attempt at a Solution

I am having trouble understanding the meaning of error δ, I tries to follow the Hint but I am not sure how comparing the power of an unknown signal would help. I calculated the Fourier transform of both interpolators but I am not sure how to continue from there. Just a clarification of what the question actually asks would be enough

 

[marcusl]

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