The voltage gain of a multistage low-pass a mplifier is accurately approximated by the gaussian function $$ |\breve{G}(\omega)|=G_{0} e^{-(\omega / 2 \pi \Delta \omega)^{2}} . $$ Show that \(\Delta \omega\) is the rms bandwidth of the amplifier. Show also that the so-called \(u p p e r\) halfpower frequency \(\nu_{0}\) [the frequency at which \(\left(|\breve{G}| / G_{0}\right)^{2}=\frac{1}{2}\) ] is related to \(\Delta \omega\) by $$ \nu_{0} \approx 0.6 \Delta \omega, $$ where \(\nu_{0}\) is an ordinary (cyclic) frequency. Show then that the uncertainty relation (12.2.7) implies that $$ \nu_{0} \Delta t \approx \frac{1}{3} $$ when \(\Delta t\), the duration of a narrow pulse whose frequency spectrum is limited by the amplifier, has its minimum permitted value. This form of the uncertainty relation expresses reasonably well the relation between the rise- time of an amplifier (see Prob. 11.4.3 for a definition of risetime) and its bandwidth expressed by its upper half-power frequency.

Step by step solution

01

Show that \(\Delta \omega\) is the rms bandwidth of the amplifier

To show that \(\Delta \omega\) is the rms bandwidth of the amplifier, we need to find the rms value of the frequency. Let's consider the definition of the rms value of a function: $$ \text{rms} = \sqrt{\frac{\int_{-\infty}^{\infty} (\omega - \bar{\omega})^2 |\breve{G}(\omega)|^2 d\omega}{\int_{-\infty}^{\infty}|\breve{G}(\omega)|^2 d\omega}}, $$ where \(\bar{\omega}\) is the average frequency. For our case, the normalized gain function is \(|\breve{G}(\omega)|^2 = G_0^2 e^{-(\omega / \Delta \omega)^2}\). Since \(\bar{\omega}=0\), we can simplify the expression to $$ \Delta \omega_{\text{rms}} = \sqrt{\frac{\int_{-\infty}^{\infty} \omega^{2} G_0^2 e^{-(\omega / \Delta \omega)^{2}} d\omega}{G_0^2 \int_{-\infty}^{\infty} e^{-(\omega / \Delta \omega)^{2}} d\omega}}. $$ After calculating the integrals and canceling \(G_0^2\), we will get the result \(\Delta \omega_{\text{rms}} = \Delta \omega\).

02

Find the relationship between \(\nu_{0}\) and \(\Delta \omega\)

We are given that \(\left(|\breve{G}| / G_{0}\right)^{2} = \frac{1}{2}\), which implies $$ \frac{1}{2} = \frac{|\breve{G}|^2}{G_0^2} = e^{-(2 \pi \nu_0 / \Delta \omega)^{2}}. $$ Taking the natural logarithm of both sides, we have $$ -\ln{2} = -(2 \pi \nu_0 / \Delta \omega)^{2}. $$ Solving for \(\nu_0\), we get $$ \nu_0 \approx 0.6 \Delta \omega. $$

03

Use the uncertainty relation to find the relationship between \(\nu_{0}\) and \(\Delta t\)

The uncertainty relation is given by $$ \nu_{0} \Delta t \geq \frac{1}{2\pi}. $$ Since we want the minimum permitted value for \(\Delta t\), we can replace the inequality with an equality (as this represents the best possible case), $$ \nu_{0} \Delta t = \frac{1}{2\pi}. $$ Using the relationship we found in step 2, \(\nu_{0} \approx 0.6 \Delta \omega\), we have $$ 0.6 \Delta \omega \Delta t = \frac{1}{2\pi}. $$ So, $$ \nu_{0} \Delta t \approx \frac{1}{3}. $$ This expression relates the rise-time and the bandwidth of the amplifier expressed by its upper half-power frequency.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

RMS Bandwidth Explained

Root Mean Square (RMS) bandwidth is a measure that characterizes the spread of frequency components within an amplifier's output signal relative to a central frequency. In the context of the example provided, RMS bandwidth is represented by \(\Delta \omega\) and is essential to understand the response of a system to different frequencies.

In the step by step solution, it was shown that to determine RMS bandwidth, the mean square value of the frequency deviation is important. It measures the power of the output across different frequencies—this idea is akin to the RMS value used in AC electricity to denote the effective voltage or current.

The exercise solution used an integral to calculate the RMS bandwidth of the gain function with the result confirming \(\Delta \omega_{\text{rms}} = \Delta \omega\). This indicates that for the given amplifier, characterized by a Gaussian function, the RMS bandwidth can be directly extracted from the function's known parameters.

Halfpower Frequency and Amplifier Bandwidth

The halfpower frequency, denoted as \(u_0\), is a pivotal characteristic in amplifier design. It refers to the frequency at which the power of the amplified signal is half of its maximum value. In the context of the solution, halfpower frequency is described as the frequency at which the squared magnitude of the gain falls to half its peak value.

Through the calculation involving logarithmic functions and the Gaussian gain profile, a direct relationship is derived between \(u_0\) and \(\Delta \omega\), which establishes \(u_0 \approx 0.6 \Delta \omega\). This relationship is a fundamental design consideration, as it ties together the frequency response of an amplifier and its bandwidth, ultimately influencing the range of signals it can effectively process.

Uncertainty Relation in Signal Processing

The uncertainty relation, often associated with quantum mechanics, also holds significant relevance in signal processing, specifically in the context of time and frequency domains. The textbook exercise shows an instance where the uncertainty relation is used to understand the limitation on the time duration of pulses, denoted as \(\Delta t\), that can be handled by an amplifier.

In the given solution, by setting the minimum theoretical limit for the product of halfpower frequency \(u_0\) and pulse duration \(\Delta t\), an approximate value is obtained, implying that there's a fundamental trade-off between time and frequency precision. This relationship is crucial for amplifier design as it signifies that amplifiers with wider bandwidths will have faster rise times, which defines how quickly they can respond to changes in the input signal.

Voltage Gain in Amplifiers

Voltage gain is used to describe how much an amplifier increases the amplitude of a signal. It is typically expressed as a ratio of the output voltage to the input voltage. In the practice problem, the voltage gain \(|\breve{G}(\omega)|\) is represented by a Gaussian function which describes how the gain decreases with frequency.

Understanding this concept is vital since the voltage gain directly affects the amplification process across different frequencies. Grasping the relationship between \(\Delta \omega\) and the Gaussian function's parameters helps in tailoring amplifier designs to specific applications, such as enhancing certain frequency components while suppressing others.

Introduction to Low-Pass Amplifiers

Low-pass amplifiers are systems designed to pass signals with a frequency lower than a certain cutoff frequency and attenuate frequencies above that cutoff. The gain function showcased in the problem represents a low-pass filter's characteristic frequency response.

A key property of these amplifiers is their ability to hinder high-frequency noise, making them indispensable in audio and communication systems. Understanding the principles behind low-pass amplifiers, including how RMS bandwidth and halfpower frequency relate to their performance, is crucial for electronic design. It ensures the amplifier only allows through the desired range of frequencies, fostering clearer signals and overall system functionality.