Problem 4 - Active Calculus

Question

In electrical engineering, a continuous function like \(f(t)=\sin t,\) where \(t\) is in seconds, is referred to as an analog signal. To digitize the signal, we sample \(f(t)\) every \(\Delta t\) seconds to form the sequence \(s_{n}=f(n \Delta t) .\) For example, sampling \(f\) every \(1 / 10\) second produces the sequence \(\sin (1 / 10), \sin (2 / 10), \sin (3 / 10), \ldots\) Suppose that the analog signal is given by $$ f(t)=(t-0.5)^{2} $$ Give the first 6 terms of a sampling of the signal every \(\Delta t=0.25\) seconds: (Enter your answer as a comma-separated list.)

Step-by-Step Solution

Answer

0.0625, 0, 0.0625, 0.25, 0.5625, 1

Step 1: Understand the function and sampling interval

The given analog signal function is \[ f(t) = (t - 0.5)^2 \]and we need to sample it every \[ \Delta t = 0.25 \] seconds.

Step 2: Calculate the first sample

To find the first term, substitute \( n = 1 \) into \[ s_n = f(n \Delta t) \]Thus, we have:\[ s_1 = f(1 \times 0.25) = f(0.25) = (0.25 - 0.5)^2 \]Simplifying:\[ s_1 = (-0.25)^2 = 0.0625 \]

Step 3: Calculate the second sample

To find the second term, substitute \( n = 2 \):\[ s_2 = f(2 \times 0.25) = f(0.5) = (0.5 - 0.5)^2 \]Simplifying:\[ s_2 = 0^2 = 0 \]

Step 4: Calculate the third sample

To find the third term, substitute \( n = 3 \):\[ s_3 = f(3 \times 0.25) = f(0.75) = (0.75 - 0.5)^2 \]Simplifying:\[ s_3 = 0.25^2 = 0.0625 \]

Step 5: Calculate the fourth sample

To find the fourth term, substitute \( n = 4 \):\[ s_4 = f(4 \times 0.25) = f(1) = (1 - 0.5)^2 \]Simplifying:\[ s_4 = 0.5^2 = 0.25 \]

Step 6: Calculate the fifth sample

To find the fifth term, substitute \( n = 5 \):\[ s_5 = f(5 \times 0.25) = f(1.25) = (1.25 - 0.5)^2 \]Simplifying:\[ s_5 = 0.75^2 = 0.5625 \]

Step 7: Calculate the sixth sample

To find the sixth term, substitute \( n = 6 \):\[ s_6 = f(6 \times 0.25) = f(1.5) = (1.5 - 0.5)^2 \]Simplifying:\[ s_6 = 1^2 = 1 \]

Key Concepts

Continuous Function

In electrical engineering, a continuous function is often used to describe an analog signal. An example of a continuous function is \(f(t) = \sin t\), where \(t\) represents time in seconds. This function represents an analog signal that varies smoothly over time. Unlike digital signals, a continuous function does not have interruptions or discrete steps. Every point in time has a corresponding value on the function's curve.

Digital Signal Processing

Digital Signal Processing (DSP) involves converting an analog signal into a digital form so it can be processed by digital systems. The first step in DSP is sampling, where the continuous analog signal is measured at regular intervals. These readings are then transformed into a sequence of numbers that represent the original signal. This digitized version can then be manipulated using algorithms to enhance, filter, or analyze the data. The process of transforming these signals is crucial for many applications, including audio processing, communications, and image manipulation.

Sequence Sampling

Sequence sampling means taking regular measurements of a continuous function to create a sequence of values. For example, if the function \(f(t) = (t - 0.5)^2\) is sampled every \(\Delta t = 0.25\) seconds, we derive values from the function at specific intervals. These values form a sequence like \(s_n = f(n \Delta t)\). This transformation allows us to analyze and process the signal more efficiently using digital tools. Each sampling point is crucial for preserving the signal's characteristics in its digital form.

Sampling Interval

The sampling interval, denoted by \(\Delta t\), refers to the time gap between each sample taken from the continuous function. For instance, in the given function \(f(t) = (t - 0.5)^2\), if \(\Delta t = 0.25\) seconds, the function is sampled every 0.25 seconds. The choice of this interval is critical; if it is too large, important information may be lost, but if it is too small, it may result in excessive data. The right balance ensures the digital representation of the signal closely resembles the original analog signal. Choosing the correct sampling interval is essential for accurate digital signal processing.