What do engineers see when they look at a real system or conceive a new design? They will look past the hardware and visualize the flow of energy, where it is stored, and where it is dissipated. They will identify the kinetic energy of moving fluids and solid masses; the potential energy of compressible fluids, elastic hoses and tanks, and deformable solids; the energy stored in electric and magnetic fields; and the energy lost when friction generates heat. But they will also see how the system might respond to real-world inputs by understanding its frequency response and the frequency spectrum of the probable inputs.

A real system often can be modeled over some range of motion and time duration as a stable, linear, time-invariant system. If the input to this system is a sine wave, the steady-state output (after the transients have died out) is a sine wave with the same frequency, but the amplitude and phase angle are both frequency dependent. Plots of the input-output amplitude ratio versus frequency and the phase angle versus frequency are called the Bode plots. If the system being excited were a nonlinear or time-varying one, the output might contain frequencies other than the input frequency, and the amplitude ratio might be dependent on the input magnitude.

Any real-world device or process will need to function properly for only a certain range of frequencies. Outside this range, we don't care what happens. When one has the frequency response curves for any system and is given a specific sinusoidal input, it is easy to calculate the sinusoidal output. What is not obvious -- but extremely important -- is that the frequency response curves are really a complete description of the system's dynamic behavior. They allow us to compute the response for any input, not just sine waves.

Jean Baptiste Fourier showed 200 years ago that any periodic waveform that exists in nature can be generated by adding up sine waves. By picking the amplitudes, frequencies, and phases of these sine waves, one can generate a waveform identical to the desired signal. A periodic function q_{i}(t) can be represented by an infinite series of terms called a Fourier series. The figures show a square wave and a plot of the first few terms of the Fourier series. The more terms one uses in the series, the better the fit will be.

Using the principle of superposition for linear systems, we can combine the frequency spectrum of a real-world signal with the system's frequency response and calculate the system time response. The device used to determine a system's frequency response experimentally is a dynamic signal analyzer, and there are many excellent application notes available on its use.

Frequency response testing plays a most significant role in grey-box modeling, which we explored in a post last month. For example, once one has the model structure for a device, such as a solenoid-operated proportional valve used in fluid power applications, the measured frequency response data can be used with optimization algorithms to determine the model parameters. And we all know that a model is worth a thousand tests. This exact situation is presented by The MathWorks.

Thanks Ratsky ... when I first read the article, I was trying to think of a mechanical example and thought of the films I'd seen of this bridge "self-destructing" but I couldn't remember details!

I second your motion! I was especially surprised to hear this coming from an ME. Both MEs and CEs should look up "Galloping Gertie," the Tacoma Narrows Bridge that failed rather spectacularly due to resonances "outside the range of interest." Also, students aren't taught about the limitations of the rote methods they learn any more. A simple example: the Fourier Transform (which underlies all frequency analysis methods) has a big qualifier: it exists ONLY for "steady-state systems" implying two things: the signal NEVER CHANGES, and perfect linearity is assumed. These are the assumptions underlying the mathemetics.

I would strongly disagree with the statement "Any real-world device or process will need to function properly for only a certain range of frequencies. Outside this range, we don't care what happens." I'd say that, in real-world systems, you'd better care about frequencies outside it "working range" or it will bite you - sometimes seriously. And in audio design, this is especially true. One glaring example is RF interference, which often arrives right along with the system input signal. Another thing this cavalier approach neglects is that ALL systems are non-linear to some degree. Now imagine the effect of two "out-of-band" (say ultrasonic frequencies for an audio system) signals applied simultaneously. The non-linearity of the system intermodulates the two and one of the results is a difference frequency that IS inside the audio range ... and audible as a non-harmonically-related "grunge" or "veil" in the audio quality. I've found that folks who approach engineering issues with only math equations (I often call them "math snobs") most frequently forget or ignore such issues. If your system can't react "gracefully" to out-of-band inputs, then one of your tasks is to remove/attenuate those frequencies so that your system doesn't "see" them! Some of the most awful-sounding audio systems are due to a DC-to-daylight bandwidth design philosophy. Any system, audio or otherwise, should include filters at the inputs. Of course, be careful what kind of filter you choose, so you don't degrade transient response (for audio, Bessel filters are the best choice because they optimize linear phase). - Bill Whitlock, president & chief engineer, Jensen Transformers www.jensen-transformers.com

I very much appreciate and agree with your approach, but the current crop of EE students has only a vague idea what you're talking about. Frequency domain analysis leads to writing and solving equations using the Laplace transform, i. e. "transfer function" equations involving ratios of polynomials in s and talking about Routh-Hurwitz, gain and phase margin (in a servo system or circuit) etc. But the current crop of students can only analyze a system expressed in the inverse z-transform, and they use an entirely different set of tools to attempt to determine the "order" of the system (an approach by the way which can quickly get you in trouble in the real world if you attempt to apply it to a system which is even marginally nonlinear). It's also true that the "order" so obtained depends in the real world on the rate at which the system is sampled (and this order is constrained to be a rather small number if the system can be "solved" at all), and hence offers very little "insight" into the principles underlying the system under analysis, and the whole process often results in the necessity to use highly complex mathematical techniques involving large mathematical matrices to make even the simplest system work properly. Nonetheless anything having to do with "frequency domain analysis" is relabeled a "legacy engineering tool" and moved to the utterly inaccessible portion of the engineering library where books are stored that were written before 1980! It strikes me that it's likely that MEs are probably taught frequency domain and EEs are taught time domain, and this could just be a scheme so professors from BOTH departmens can get consulting assignments out of the disparity (!), anyway I'd like to hear your comments because I believe BOTH methods are useful for solving "real world" problems.

Thanks for the informative post. Frequency response plays a major role in audio design and development. It can be to study the response of the audio components to any frequency.

Actually in real world, no frequencies should be exaggerated or reduced, more accurate representation of the original sound is required. Any audio device should preserve the loudness relationship between various instruments and voices and should not over or under-emphasize any frequency. This can be achieved if frequency response of device is flat. A flat frequency response means that the audio device is equally sensitive to all frequencies.

More often than not, with the purchase of a sports car comes the sacrifice of any sort of utility. In other words, you can forget about a large trunk, extra seats for the kids, and more importantly driving in snowy (or inclement) weather. But what if there was a vehicle that offered the best of both worlds; great handling and practicality?

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