Electrical Engineering - reconstruction filter, what is the frequency and output?

Question

could someone please help me with the following DSP question.

If the DSP block in Figure 1 is assumed to have a sampling frequency of 25 kHz and performs the following difference equation: y[n] = x[n], (that is, the output of the A2D is copied directly to the input of the D2A) what is the frequency and approximate amplitude of the signal, y(t), observed at the output of the reconstruction filter, when:

a. x(t) is a sinusoidal signal of frequency 10 kHz;

b. x(t) is a sinusoidal signal of frequency 12.5 kHz;

c. x(t) is a sinusoidal signal of frequency 15 kHz;

d. x(t) is a sinusoidal signal of frequency 25 kHz;

e. x(t) is a square wave of fundamental frequency 4 kHz;

where figure 1 is a block diagram of the following

x(t)->anti-aliasing filter->S/H->Q->DSP->D/A->reconstruction filter->y(t)

Any help would be appreciated, thanks.

Answer 1

nyquist /sahnnon  theory says if a  Band limited  signal  sampled twice of its highest freq content rate then it can be  faithfully  reconstructed.

Since your S/H is working at 25 KHZ  hence it can only reconstruct the signal at

a. &b correctly

while  c&d will be stepped up signal being under sampled .

e. signal e would be over sampled signal.

Answer 2

Ok, you don;t say much that would allow a full answer.

Assumptions:

Q is a quantizer & an appropriate one, ie enough dynamic range (bits if you like)

Q is linear

S&H similar assumption.

D/A similar assumptions

antialias filter is "perfect" in some way & appropriate for 25kHz sample rate.

Reconstruction filter is also "perfect" in some ways & appropriate for 25kHz sample rate.

Assume baseband sampling: refer later.

So the question is likely about aliasing & appropriate sample rates.

10kH sampled at 25kHz: 10kHz s less than the "Nyquist frequency" so it gets more than two samples per cycle & comes out of a perfect reconstruction filter unaltered.

12.5kHz is exactly the Nyquist frequency. So interesting.

If we happen to sample when the signal is at its positive peak, then the  next sample will be at its negative peak & so on.  Perfect reconstruction will provide a full amplitude sin.

Any real anti-alias filter or reconstruction filter will roll off before then & result in attenuation: smaller signal.

If by chance the sampling were to fall on the place that the signal passes through zero, then all fall exactly so & the output is zero (DC).

All other cases of sin 12.5kHz fall somewhere between these: lower amplitude, 12.5kHz sin waves.

15kHz is greater than the Nyquist & hence will look like (15kHz - 25kHz). Yes technically negative frequency. So the sin you get out will look like 10kHz.  The sample will fall on values indistinquishable & therefore reconstruct as a 10kHz sin wave.

If the perfect antialias filter is appropriate for 25kHz sample, ie blocks everything above 12.5kHz, then this 15kHz likely does not get through & nothing comes out.

If the antialias filter is not perfect, then likely something comes out at teh 10kHz, but attenuated by the amount the antialias filter cuts it down.

25kHz: right on the sample rate.

Wherever the first sample falls, all other samples will fall exactly an even number of samples later. Hence DC out.  The value depends on the fluke of where the first sample falls, ie the relative phase of the sampling & signal. That is it is indistinguishable from (25kHz - 25kHz) = 0Hz.

Again, though this likely does not get through the antialias filter: hence nothing arrives at the ADC.

square wave: you can work it out. Clearly the fundamental will get through.  Square wave has next component at 1/3 amplitude 3rd hamonic. ie 12 kHz, etc.

Potentially the antialias filter knocks off harmonics rounding it off, the sampler aliases the remaining frequencies & its a little complicated what remains.  You can calculate the first few harmonics & assume the higher ones are attenuated to inconsequential.

Or you can work out where the sample fall & sort it out from there.

BUT: this is not the only answer. (depending on your level & interest)

If for example one knows that the input signal is round 25kHz rather than DC, then one could design the antialias filter & reconstruction filter for this bandpass situation.  In "bandpass sampling" it is the bandwidth of the signal that determines the sampling rate, not the highest frequency.