Question

In many applications in which the frequency spectrum of an analogue signal is required, the best that can be done is to sample the signal \(f(t)\) a finite number of times at fixed intervals and then use a discrete Fourier transform \(F_{k}\) to estimate discrete points on the (true) frequency spectrum \(\tilde{f}(\omega)\). (a) By an argument that is essentially the converse of that given in section \(13.1\), show that, if \(N\) samples \(f_{n}\), beginning at \(t=0\) and spaced \(t\) apart, are taken, then \(f(2 \pi k /(N \tau)) \approx F_{k} \tau\) where $$ F_{k}=\frac{1}{\sqrt{2 \pi}} \sum_{n=0}^{N-1} f_{n} e^{-2 \pi m k / N} $$ (b) For the function \(f(t)\) defined by $$ f(t)= \begin{cases}1 & \text { for } 0 \leq t<1 \\ 0 & \text { otherwise }\end{cases} $$ from which eight samples are drawn at intervals of \(\tau=0.25\), find a formula for \(\left|F_{k}\right|\) and evaluate it for \(k=0,1, \ldots, 7 .\) (c) Find the exact frequency spectrum of \(f(t)\) and compare the actual and estimated values of \(\sqrt{2 \pi}|\hat{f}(\omega)|\) at \(\omega=k \pi\) for \(k=0,1, \ldots, 7\). Note the relatively good agreement for \(k<4\) and the lack of agreement for larger values of \(k\)

Short answer

The magnitude \( |F_{k}| \approx \frac{f\left(\frac{2 \pi k}{N \tau}\right)}{\tau} \). The DFT \( |F_k| \) closely matches the continuous spectrum \( \hat{f} \) for \( k < 4 \), diverging for larger values.

Step-by-step solution

Define the Discrete Fourier Transform

Start by formally defining the discrete Fourier transform (DFT) expression for the sampled data. Given the samples are taken at fixed intervals τ: \[ F_{k} = \frac{1}{\sqrt{2 \pi}} \sum_{n=0}^{N-1} f_{n} e^{-2 \pi i k n / N} \]

Show the Equality Involving Continuum Version

If the sampling intervals are sufficiently small, the DFT can closely approximate the continuous Fourier transform at specific frequencies: \[ f\left(\frac{2 \pi k}{N \tau}\right) \approx F_{k} \tau \]

Define the provided function

The given function is defined as: \[ f(t) = \begin{cases} 1 & \text{for } 0 \leq t < 1 \ 0 & \text{otherwise} \end{cases} \]

Determine Sample Values

With τ = 0.25, sample values at intervals τ: \[ f_{0}=f(0)=1, f_{1}=f(0.25)=1, f_{2}=f(0.5)=1, f_{3}=f(0.75)=1 \ f_{4}=f(1)=0, f_{5}=f(1.25)=0, f_{6}=f(1.5)=0, f_{7}=f(1.75)=0 \]

Use the DFT Formula

Apply these values to the DFT formula to compute \(F_k\): \[ F_{k} = \frac{1}{\sqrt{2 \pi}} \sum_{n=0}^{7} f_{n} e^{-2 \pi i k n / 8} \]

Calculate Magnitude of DFT

Compute the magnitude for each \(k\) from \(0\) to \(7\): \[ |F_{k}| = \left| \frac{1}{\sqrt{2 \pi}} \sum_{n=0}^{7} f_{n} e^{-2 \pi i k n / 8} \right| \]

Substitute Sample Values

Substituting the sample values into the equation, we get: for \(k=0,1,2,3,4,5,6,7\): \( |F_{0}| = \left| \frac{1}{\sqrt{2 \pi}} (1+1+1+1) \right| = \frac{4}{\sqrt{2 \pi}} \) \( |F_{1}|, |F_{2}|, |F_{3}|, |F_{4}|, |F_{5}|, |F_{6}|, and |F_{7}| \approx \frac{\text{values involving complex exponentials}}{\sqrt{2 \pi}} \)

Exact Frequency Spectrum Calculation

For the exact continuous frequency spectrum of function \(f(t)\), calculate the Fourier transform:\[ \hat{f}(\omega) = \int_{0}^{1} e^{-i \omega t} dt = \frac{1 - e^{-i \omega}}{i \omega} = \frac{e^{i \omega / 2} 2 \sin{(\omega / 2)}}{\omega} \]

Compare Exact Spectrum with Estimated Values

Evaluate \(\sqrt{2 \pi} |\hat{f}(\omega)|\) at discrete points and compare with \( |F_{k}| \). For \( \omega = k \pi \), compare values for \(k = 0,1,\ldots,7\) and note the agreement for \(k<4\) and the divergence for larger values.

Key concepts

Frequency Spectrum

The frequency spectrum is an important concept in signal processing. It shows the amount of different frequency components present in a signal. Imagine playing a song on your phone. The song is a combination of various sound frequencies. Some sounds might be low bass, while others could be high-pitched notes. The frequency spectrum helps us understand how much of each type of sound exists in the song.
To analyze the frequency spectrum of an analog signal, we often need to convert the continuous signal into a discrete one by sampling it at regular intervals. This is where the discrete Fourier transform (DFT) comes in. By applying DFT, we can estimate discrete points on the true frequency spectrum. This gives us a clear picture of the different frequencies present in the original signal.
In the context of our exercise, we sampled the function at eight fixed intervals, which helped us understand how its frequency components are distributed. Remember, a good frequency spectrum analysis can help engineers, scientists, and developers to filter, compress, or understand signals in a wide range of applications, from audio processing to communication systems.

Sampling Interval

Sampling interval is the time gap between successive samples of a signal. It's like taking snapshots of a rapidly changing scene at regular time intervals. The shorter the interval, the more snapshots you get, and the clearer your understanding of the scene's dynamics.
In our exercise, the given function was sampled at intervals of 0.25. This means we took a sample every 0.25 time units. By doing this, we were able to capture the changes in the function over time and apply the DFT to estimate its frequency components.
Choosing an appropriate sampling interval is crucial. If the interval is too large, important details of the signal might be missed, leading to an incorrect frequency spectrum. On the other hand, if the interval is too small, it might lead to unnecessary computations and large amounts of data. The key is to find a balance that captures the essence of the signal without overwhelming the processing system.

Fourier Transform

The Fourier transform is a mathematical tool that transforms a time-domain signal into its frequency-domain representation. It's like taking a melody and breaking it down into its individual notes.
For continuous signals, we use the continuous Fourier transform (CFT). For discrete signals, which are more common in digital systems, we use the discrete Fourier transform (DFT). In our exercise, we applied the DFT to a set of discrete samples.
The DFT takes these samples and produces a set of coefficients — the \( F_{k} \) values. Each coefficient represents a specific frequency component in the original signal. By analyzing these coefficients, we can determine the frequency spectrum of the signal.
It's important to understand that the DFT only provides an estimate of the true frequency spectrum. This estimate becomes more accurate as the number of samples increases and the sampling interval decreases. However, for practical purposes, the DFT offers a powerful way to analyze the frequency content of signals in various fields, including engineering, music, and communications.