Sampling a continuous-time signal, aliasing/Nyquist

[alc95]

Homework Statement

The analog signal x(t) = cos(2pi f t) is sampled at a rate of 1 kHz, using ideal
impulse sampling, to obtain the sampled signal x^(s)(t). The sampled signal is then sent through an ideal
lowpass filter with transfer function H(2pi f ) = 0.001 rect (0.001 f ).
(a) If f =1.01kHz, what is the output frequency from the filter? Is there aliasing or not?
(b) If f = 0.99kHz, what is the output frequency from the filter? Is there aliasing or not?
(c) If f = 0.49kHz, what is the output frequency from the filter? Is there aliasing or not?

Homework Equations

To perfectly reconstruct a signal, we need: sampling rate > 2*signal frequency

The Attempt at a Solution

I think these are correct, not 100% sure though:
(a)
aliasing occurs, as 1.00 kHz !> 2 × 1.01 kHz
fout = 1.01 kHz – n × 1.00 kHz = 1.01 kHz – 1 × 1.00 kHz = 0.01 kHz

(b)
aliasing occurs, as 1.00 kHz !> 2 × 0.99 kHz
fout = 0.99 kHz – n × 1.00 kHz = 0.99 kHz – 1.00 kHz = - 0.01 kHz

(c)
aliasing does not occur, as 1.00 kHz > 2 × 0.49 kHz
fout = 0.49 kHz

 

[rude man]

Homework Statement

The analog signal x(t) = cos(2pi f t) is sampled at a rate of 1 kHz, using ideal
impulse sampling, to obtain the sampled signal x^(s)(t). The sampled signal is then sent through an ideal
lowpass filter with transfer function H(2pi f ) = 0.001 rect (0.001 f ).

Not sure I understand this terminology. What really is the ideal low-pass cutoff frequency? Is it really a function of input frequency f? Do you have to divide by 2π? Etc. ?

I don't know your method, and as I say I'm not sure what your cutoff filter really looks like, but it basically seems to work.

Just FYI I first determine the cutoff frequency of the low-pass filter = f₀, the I take the sampling frequency f_s and my input frequency f and find spectral components:

f, |f_s - f|, f_s + f, |2f_s - f|, 2f_s + f, ...

and then compare each component against the cutoff frequency f₀.
The only unaliased signal is at f so anything below f₀ that is a mix of f and f_s is an aliased component.