answer:Given a band-limited signal x(t)x(t) and an ideal sampling function s(t)s(t) with a frequency twice that of the band-limited signal, we can express the spectrum of the sampled signal mathematically. The ideal sampling function can be represented as a Dirac Comb: s(t)=∑n=−∞∞δ(t−nTs)s(t)=∑n=−∞∞δ(t−nTs)where TsTs is the sampling period, and fs=1Tsfs=1Ts is the sampling frequency. The sampled signal xs(t)xs(t) is obtained by multiplying the continuous-time signal x(t)x(t) with the sampling function s(t)s(t): xs(t)=x(t)s(t)=x(t)∑n=−∞∞δ(t−nTs)xs(t)=x(t)s(t)=x(t)∑n=−∞∞δ(t−nTs)To find the spectrum of the sampled signal, we need to compute the Fourier transform of xs(t)xs(t). We can use the convolution property of Fourier transforms, which states that the Fourier transform of the product of two functions is equal to the convolution of their Fourier transforms: Xs(f)=xs(t)=x(t)s(t)=X(f)∗S(f)Xs(f)=Fxs(t)=Fx(t)s(t)=X(f)∗S(f)Here, X(f)X(f) is the Fourier transform of the band-limited signal x(t)x(t), S(f)S(f) is the Fourier transform of the sampling function s(t)s(t), and ∗∗ denotes convolution. The Fourier transform of the Dirac Comb is another Dirac Comb: S(f)=∑k=−∞∞δ(f−kfs)S(f)=∑k=−∞∞δ(f−kfs)Now, we can find the spectrum of the sampled signal by convolving X(f)X(f) with S(f)S(f): Xs(f)=X(f)∗∑k=−∞∞δ(f−kfs)Xs(f)=X(f)∗∑k=−∞∞δ(f−kfs)By the sifting property of the Dirac delta function, the convolution results in shifting the spectrum of X(f)X(f) to the center frequencies of each impulse in S(f)S(f): Xs(f)=∑k=−∞∞X(f−kfs)Xs(f)=∑k=−∞∞X(f−kfs)This expression represents the spectrum of the sampled signal from a purely mathematical point of view. The spectrum consists of infinite replicas of the original signal spectrum, shifted by integer multiples of the sampling frequency fsfs.

answer:As we increase the sampling frequency fsfs, the sampling period Ts=1fsTs=1fs decreases. This has an impact on the spectrum of the sampled signal Xs(f)Xs(f). Recall that the spectrum of the sampled signal is given by: Xs(f)=∑k=−∞∞X(f−kfs)Xs(f)=∑k=−∞∞X(f−kfs)As we increase the sampling frequency, the distance between the replicas of the original signal spectrum in the frequency domain increases. This is because the replicas are centered at integer multiples of the sampling frequency, kfskfs. With a higher sampling frequency, the replicas of the signal spectrum are further apart from each other. This reduces the risk of overlapping or aliasing between adjacent replicas. When the replicas do not overlap, it is possible to perfectly reconstruct the original band-limited signal from the sampled signal using an ideal low-pass filter with a cutoff frequency equal to half of the increased sampling frequency. In summary, as the sampling frequency increases, the distance between the replicas of the signal spectrum in the frequency domain increases, reducing the risk of aliasing and making it easier to recover the original signal using an ideal low-pass filter.

answer:To determine the value of the strong coupling constant (αs) at a center-of-mass energy of 91.2 GeV, we need the experimental measurements of hadronic cross-sections at the Z^0 peak. However, you have not provided any specific measurements or data. In general, the strong coupling constant can be extracted from the hadronic cross-section measurements using the following relation: σ_had = σ_0 * (1 + αs/π + ...) where σ_had is the hadronic cross-section, σ_0 is the lowest-order cross-section, and the terms in the parentheses represent the perturbative QCD corrections, with αs being the strong coupling constant. Once you provide the experimental measurements of hadronic cross-sections, we can proceed to calculate the value of αs at a center-of-mass energy of 91.2 GeV.

answer:Question: Implement a simple rate limiter using Node.js. Problem: Design a rate limiter middleware function for an Express.js application that restricts the number of requests a client can make within a specified time window. The rate limiter should have the following requirements: 1. Allow a maximum of `n` requests per client within a time window of `t` seconds. 2. After exceeding the allowed requests within the time window, respond with a 429 (Too Many Requests) status code and an appropriate error message. 3. The rate limiter should use the client's IP address to track requests. Function signature: ```javascript function rateLimiter(maxRequests, timeWindowInSeconds) { // Your implementation goes here } ``` Example: If `maxRequests` is set to 5 and `timeWindowInSeconds` is set to 60, the rate limiter should allow a maximum of 5 requests per client within a 60-second time window. Any subsequent requests within this time window should result in a 429 status code response. Hints: 1. Consider using the `req.ip` property to get the client's IP address. 2. You may use JavaScript's `Map` to store the IP addresses and their request information. 3. Make use of `setTimeout` to clear the stored request data when the time window expires.