Frequency Response of Amplifiers. Concept of frequency response, Human ear response to audio frequencies, significance of Octaves and Decades. The decibel unit. Square wave testing of amplifiers, Miller's theorem. Effect of coupling, bypass, junction and stray capacitances on frequency response for BJT and FET amplifiers. Concept of dominant pole.
Frequency response requirements differ depending on the application.Inhigh fidelityaudio, an amplifier requires a frequency response of at least 20–20,000 Hz, with a tolerance as tight as ±0.1 dBin the mid-range frequencies around 1000 Hz, however, intelephony, a frequency response of 400–4,000 Hz, with a tolerance of ±1 dB is sufficient for intelligibility of speech.
Frequency response curves are often used to indicate the accuracy of electronic components or systems.When a system or component reproduces all desired input signals with no emphasis or attenuation of a particular frequency band, the system or component is said to be "flat", or to have a flat frequency response curve.
Frequency response of a low pass filter with 6 dB per octave or 20 dB per decade
The frequency response is typically characterized by themagnitudeof the system's response, measured in decibels (dB), and thephase, measured inradians, versus frequency. The frequency response of a system can be measured by applying atest signal, for example:
§applying an impulse to the system and measuring its response (seeimpulse response)
§sweeping a constant-amplitude pure tone through thebandwidthof interest and measuring the output level and phase shift relative to the input
These typical response measurements can be plotted in two ways: by plotting the magnitude and phase measurements to obtain aBode plotor by plotting the imaginary part of the frequency response against the real part of the frequency response to obtain aNyquist plot.
Once a frequency response has been measured (e.g., as an impulse response), providing the system islinear and time-invariant, its characteristic can be approximated with arbitrary accuracy by adigital filter. Similarly, if a system is demonstrated to have a poor frequency response, a digital oranalog filtercan be applied to the signals prior to their reproduction to compensate for these deficiencies.
Frequency response measurements can be used directly to quantify system performance and design control systems. However, frequency response analysis is not suggested if the system has slow dynamics.
Thefrequency responseof anLTI filtermay be defined as thespectrumof the outputsignaldivided by thespectrumof the input signal. In this section, we show that the frequency response of anyLTIfilteris given by itstransfer functionevaluated on the unit circle,i.e.,. We then show that this is the same result we got usingsine-waveanalysis in Chapter1.
Thus, the spectrum of the filter output is just the input spectrum times the spectrum of the impulse response. We have therefore shown the following:
This immediately implies the following:
We can express this mathematically by writing
By Eq.(7.2), the frequency response specifies thegainandphase shiftapplied by the filter at each frequency. Since,, andare constants, the frequency responseis only a function of radian frequency. Sinceis real, the frequency response may be considered acomplex-valued function of a real variable. The response at frequencyHz, for example, is, whereis thesampling periodin seconds. It might be more convenient to define new functions such asand write simplyinstead of having to writeso often, but doing so would add a lot of new functions to an already notation-rich scenario. Furthermore, writingmakes explicit the connection between the transfer function and the frequency response.
Notice that defining the frequency response as a function ofplaces the frequency ``axis'' on theunit circlein the complexplane, since. As a result, adding multiples of thesamplingfrequency tocorresponds to traversing whole cycles around the unit circle, since
whenever is an integer. Since every discrete-time spectrum repeats in frequency with a ``period'' equal to the sampling rate, we may restrict to one traversal of the unit circle; a typical choice is [ ]. For convenience, is often allowed.
We have seen that the spectrum is a particular slice through the transfer function. It is also possible to go the other way and generalize the spectrum (defined only over the unit circle) to the entireplane by means ofanalytic continuation(§D.2). Since analytic continuation is unique (for all filters encountered in practice), we get the same result going either direction.
Because everycomplex numbercan be represented as a magnitudeand angle,viz.,, the frequency responsemay be decomposed into two real-valued functions, theamplitude responseand thephase response. Formally, we may define them as follows:
Frequency Response Ranges
You will often see frequency response quoted as a range between two figures. This is a simple (or perhaps "simplistic") way to see which frequencies a microphone is capable of capturing effectively. For example, a microphone which is said to have a frequency response of 20 Hz to 20 kHz can reproduce all frequencies within this range. Frequencies outside this range will be reproduced to a much lesser extent or not at all.
This specification makes no mention of the response curve, or how successfully the various frequencies will be reproduced. Like many specifications, it should be taken as a guide only.