Copy the screen display on the spectrum analyzer to include in your report. Labe

Question

Copy the screen display on the spectrum analyzer to include in your report. Label the graph.

Frequency of the signal to be sampled:

Sampling frequency:

Is the sampling theorem satisfied?

List frequencies of the sampled signals in the range from 0 to 10 kHz

Frequencies:

a. The original signal cannot be fully recovered by anti-image filtering the sampled signal if the sampling condition is not satisfied.

Left double click on the second spectrum analyzer attached to the anti-imaging filter.
Run the simulation using the same setting of the spectrum analyzer

Copy the screen display on the spectrum analyzer to include in your report. Label the graph.

Do you fully recover the original signal?

List the aliasing frequency if any:

Check the oscilloscope for displaying signals in time domain. (Capture the screen to include in your report.)


b. Now using the same setting for the sinusoidal function as the following
Frequency = 7000 Hz
Vp (amplitude) =1 volts=0.707 rms,
DC offset = 1 volt
Connect the sinusoidal function output to the input of the anti-aliasing filter as in Figure 3.
Run the simulation using the same setting for both the spectrum analyzers.

Frequency of the signal to be sampled:

Sampling frequency:

Is the sampling theorem satisfied?

Can you find frequencies of the sampled signals for the range from 0 to 10 kHz?
From the first spectrum analyzer:

Explanation / Answer

The Nyquist–Shannon sampling theorem, after Harry Nyquist and Claude Shannon, is a fundamental result in the field of information theory, in particular telecommunications and signal processing. Sampling is the process of converting a signal (for example, a function of continuous time or space) into a numeric sequence (a function of discrete time or space). Shannon's version of the theorem states:[1] If a function x(t) contains no frequencies higher than B hertz, it is completely determined by giving its ordinates at a series of points spaced 1/(2B) seconds apart. The theorem is commonly called the Nyquist sampling theorem; since it was also discovered independently by E. T. Whittaker, by Vladimir Kotelnikov, and by others, it is also known as Nyquist–Shannon–Kotelnikov, Whittaker–Shannon–Kotelnikov, Whittaker–Nyquist–Kotelnikov–Shannon, WKS, etc., sampling theorem, as well as the Cardinal Theorem of Interpolation Theory. It is often referred to simply as the sampling theorem. In essence, the theorem shows that a bandlimited analog signal can be perfectly reconstructed from an infinite sequence of samples if the sampling rate exceeds 2B samples per second, where B is the highest frequency of the original signal. If a signal contains a component at exactly B hertz, then samples spaced at exactly 1/(2B) seconds do not completely determine the signal, Shannon's statement notwithstanding. This sufficient condition can be weakened, as discussed at Sampling of non-baseband signals below. More recent statements of the theorem are sometimes careful to exclude the equality condition; that is, the condition is if x(t) contains no frequencies higher than or equal to B; this condition is equivalent to Shannon's except when the function includes a steady sinusoidal component at exactly frequency B. The theorem assumes an idealization of any real-world situation, as it only applies to signals that are sampled for infinite time; any time-limited x(t) cannot be perfectly bandlimited. Perfect reconstruction is mathematically possible for the idealized model but only an approximation for real-world signals and sampling techniques, albeit in practice often a very good one. The theorem also leads to a formula for reconstruction of the original signal. The constructive proof of the theorem leads to an understanding of the aliasing that can occur when a sampling system does not satisfy the conditions of the theorem. The sampling theorem provides a sufficient condition, but not a necessary one, for perfect reconstruction. The field of compressed sensing provides a stricter sampling condition when the underlying signal is known to be sparse. Compressed sensing specifically yields a sub-Nyquist sampling criterion. A signal or function is bandlimited if it contains no energy at frequencies higher than some bandlimit or bandwidth B. The sampling theorem asserts that, given such a bandlimited signal, the uniformly spaced discrete samples are a complete representation of the signal as long as the sampling rate is larger than twice the bandwidth B. To formalize these concepts, let x(t) represent a continuous-time signal and X(f) be the continuous Fourier transform of that signal: X(f) stackrel{mathrm{def}}{=} int_{-infty}^{infty} x(t) e^{- i 2 pi f t} { m d}t. The signal x(t) is said to be bandlimited to a one-sided baseband bandwidth, B, if X(f) = 0 quad for all |f| > B,, or, equivalently, supp(X) ? [-B, B].[2] Then the sufficient condition for exact reconstructability from samples at a uniform sampling rate fs (in samples per unit time) is: f_s > 2 B.! The quantity 2B is called the Nyquist rate and is a property of the bandlimited signal, while fs/2 is called the Nyquist frequency and is a property of this sampling system. The time interval between successive samples is referred to as the sampling interval: T stackrel{mathrm{def}}{=} rac{1}{f_s},, and the samples of x(t) are denoted by x(nT) for integer values of n. The sampling theorem leads to a procedure for reconstructing the original x(t) from the samples and states sufficient conditions for such a reconstruction to be exact. when there is no overlap of the copies (aka "images") of X(f), the k = 0 term of Xs(f) can be recovered by the product: X(f) = H(f) cdot X_s(f),, where: H(f) = egin{cases}1 & |f| f_s - B. end{cases} H(f) need not be precisely defined in the region [B, fs - B] because Xs(f) is zero in that region. However, the worst case is when B = fs/2, the Nyquist frequency. A function that is sufficient for that and all less severe cases is: H(f) = mathrm{rect} left(rac{f}{f_s} ight) = egin{cases}1 & |f| < rac{f_s}{2} \ 0 & |f| > rac{f_s}{2}, end{cases} where rect(u) is the rectangular function. Therefore: X(f) = mathrm{rect} left(rac{f}{f_s} ight) cdot X_s(f) = mathrm{rect} (Tf) cdot T sum_{n=-infty}^{infty} x(nT) e^{-i 2pi n T f} (from Eq.1, above). = T sum_{n=-infty}^{infty} x(nT)cdot mathrm{rect} (Tf) cdot e^{-i 2pi n T f}. The original function that was sampled can be recovered by an inverse Fourier transform: x(t) = mathcal{F}^{-1}left { T sum_{n=-infty}^{infty} x(nT)cdot mathrm{rect} (Tf) cdot e^{-i 2pi n T f} ight} = T sum_{n=-infty}^{infty} x(nT)cdot underbrace{mathcal{F}^{-1}left { mathrm{rect}(Tf) cdot e^{-i 2pi n T f} ight}}_{rac{1}{T}cdot mathrm{sinc} left( rac{t - nT}{T} ight)} [3] = sum_{n=-infty}^{infty} x(nT)cdot mathrm{sinc} left( rac{t - nT}{T} ight), which is the Whittaker–Shannon interpolation formula. It shows explicitly how the samples, x(nT), can be combined to reconstruct x(t). From Figure 8, it is clear that larger-than-necessary values of fs (smaller values of T), called oversampling, have no effect on the outcome of the reconstruction and have the benefit of leaving room for a transition band in which H(f) is free to take intermediate values. Undersampling, which causes aliasing, is not in general a reversible operation. Theoretically, the interpolation formula can be implemented as a low pass filter, whose impulse response is sinc(t/T) and whose input is extstylesum_{n=-infty}^{infty} x(nT)cdot delta(t - nT), which is a Dirac comb function modulated by the signal samples. Practical digital-to-analog converters (DAC) implement an approximation like the zero-order hold. In that case, oversampling can reduce the approximation error.