Aperture and aliasing; sampling 10cos(20πt)cos(200πt) at 250 samples/s

Aperture effect

In flat-top sampling each sample is held for a width $\tau$, which is the same as passing ideal samples through a rectangular pulse filter $H(f) = \tau\,\text{sinc}(f\tau)e^{-j\pi f\tau}$. The sampled spectrum $f_sH(f)\sum_n G(f - nf_s)$ is therefore shaped by a sinc that falls with frequency, so the higher message frequencies are attenuated. This amplitude distortion is the aperture effect. It is made small by keeping $\tau \ll T_s$ and corrected by an equalizer with response $1/H(f)$.

Aliasing effect

If the sampling rate is below the Nyquist rate ($f_s < 2W$), or the signal is not band-limited, the shifted copies of the spectrum overlap. Components above $f_s/2$ fold back and appear as false lower frequencies, and the original signal cannot be recovered. It is prevented by an anti-aliasing LPF before the sampler and by sampling above $2W$.

Numerical

$$g(t) = 10\cos(20\pi t)\cos(200\pi t) = 5\cos(180\pi t) + 5\cos(220\pi t)$$

So $g(t)$ has two tones: 90 Hz and 110 Hz, each of amplitude 5. Highest frequency $W = 110$ Hz.

(i) Spectrum of the sampled signal ($f_s = 250$ Hz)

$$\begin{aligned} G(f) &= 2.5\big[\delta(f \pm 90) + \delta(f \pm 110)\big] \\ G_\delta(f) &= f_s\sum_n G(f - nf_s) = 625\sum_n\big[\delta(f - 250n \pm 90) + \delta(f - 250n \pm 110)\big] \end{aligned}$$

Each line has weight $250 \times 2.5 = 625$. Positive-frequency lines:

 625 |  | |  | |      | |  | |
     +--+-+--+-+------+-+--+-+--> f (Hz)
       90 110 140 160  340 360 ...
          (baseband)  (around 250)

The baseband pair (90, 110 Hz) and the lowest image pair (140, 160 Hz) do not overlap, since $f_s = 250 > 2W = 220$.

(ii) Reconstruction filter cut-off An ideal LPF must pass 110 Hz and reject 140 Hz:

$$110\ \text{Hz} < f_c < 140\ \text{Hz}$$

A convenient choice is the midpoint $f_c = f_s/2 = 125$ Hz.

(iii) Nyquist rate

$$f_N = 2W = 2 \times 110 = 220\ \text{Hz}$$

Answer: lines at $250n \pm 90$ and $250n \pm 110$ Hz (weight 625 each); cut-off between 110 and 140 Hz (e.g. 125 Hz); Nyquist rate = 220 Hz.