Yes, if you can find Y(s) and U(s) by the Laplace transform applied to u(t) and y(t). Should the frequency response be bandlimited depends in the signals. That what is called the system identification problem.

Since u(t) and y(t) are signals that are obtained from sensors, they are only a series of samples. Hence, deriving U(s) and Y(s) is non-trivial. If you know any technique for getting Y(s) (or Y(z)) given a sequence y(t) (or y(k)), I would be glad to know that.

In my case, I have an apparatus whose frequency response is of interest, but, its model (i.e., system identification) is not of interest.

Is it possible to use Frequency Response Function ?

It is simplier for implementation, because it is needs only FFT.

Moreover there is well known H1, H2 (or H3) FRF funcions for systems with hum from sensors.

What about this

you have discrete samples of u(t) and y(t), use simple convolution operation and come up with h(t) by applying the results in reverse order

h=u*y, implement using convolution sum operation y[n]=SUM (u[k].h[n-k])

you can assume few samples in delay line and can put zeros where needed and can refine the h(t). Later simple fft can be applied to get H(f)

Also, If system is assumed to be LTI and if you are able to pass complex sinusoid exp (jw0) from the system. The output shall contain the same signal coupled with system's frequency response as these input signals are the eigen function of linear systems that can be aproximated by some technique

Another approach can be if you can pass a spike at some time and measure the output that would be aproximated to be h(t) of the system and can be tranformed to H(f) through fft.

Janardhanan, if you have a series of samples, then solve the same problem in the discrete-time domain employing the discrete Fourier transform or z-transform. Anyway, to identify a system means to ascertain one of its general functions, such as the transfer function, frequency response or impulse response via the input and output.

Vibration testing uses this approach all the time. True, the fact that your signal are band-limited limits the validity of your model to the frequency regime where your excitation (u(t)), has some energy. There are several methods that allow you to find the frequency response. The most suitable method depends on the system and your aim. If the system has no resonance peaks, you can fit a time-series based model (i.e. ARMA, OE, see book by L. Ljung Model Identification). Having obtained a model, in this case the coefficients of the difference equation of differential equation, you can compute the frequency response. Other methods may use a non-parametric approach whereby you make use of the Discrete Fourier Transform (DFT) in its efficient form FFT to produce the frequency response. There are some guidelines to perform this task correctly (e.g. Bendat & Piersol http://www.amazon.com/Random-Data-Measurement-Procedures-Probability/dp/0470248777 ).

The short answer: use existing codes like the function etfe in Matlab.

solving the problem with DFT in time domain -discrete will be a good approach to study about the whole properties of the discrete system.

If you have sampling rate of your signal, you can obtain its frequency response by Matlab software easily. I can help you if you need. my thesis is on signal processing and these problem are commonplace in this field.

I agree with Yuriy Shmaliy. In the discrete-time domain, DFT or z-transform is a effective signal analysis tool.

I agree with Imran Shafi . you can refer to Matlab tools to obtain fft from your transient response , and you can use some other functions in robust control toolbox to deduce some analysis of the system based on your frequency response

I think I should have been much more clear with the way in which the question is phrased.

In my problem, I am trying to design a controller based on the response of the system without assuming any particular model or model structure. Hence, the emphasis is on getting the frequency response without the help of an explicit model or model structure.

I am well aware of the system identification toolbox in matlab and I know that a model and consequently a frequency response can be obtained IF a model structure is assumed. This assumption is not to be made in my problem.

In my pursuit to an answer to this, I have tried all the toolboxes of MATLAB and was surprised that there is no answer to this question.

I had posted it here, in the forum of researchers so that somebody can shed light on "How to get the frequency response from the time response data samples, without assuming a model structure'.

You might not have a model in hand, that does not mean that your system does not have a model. What every one here is suggesting is to deconvolve your output from the input which essentially will give you the transfer function and thereby your model. Also as many suggested here you would have to band limit your signal. As long as you sample the output and input at the same rate there are many ways to do it.

Ravi, of course the system has a model. But, what is the order to be chosen? Are there dead times in some of the subsystems? Does the system have an integral effect on the input? These are the assumptions that you need to make before finding the model. Incorrect assumptions lead to incorrect results.

Previous answer by Ravi Seethamraju is correct. In system Identification, it is called the nonparametric approach. It is described in standard book on identification such as Ljung's one or Soderstrom and Stoica one and it is linked to spectral estimation or Fourier analysis. it is implemented in the matlab toolbox system identification

Yes, you can. You hinted at the response already. Laplace transform is the general mathematical framework. Fourier transform is a special case of that. Go to the library.

The problem is that I have neither U(s), nor Y(s), nor any analytical representation of u(t) or y(t).

@Janardhanan Sivaramakrishnan, you don't need the analytical representation of input and output, just their samples. Go to the library. Any good book on digital signal processing will help.

Best of luck.

Usual nonparametric system identification methods using the Fourier transform, Fourier analysis cannot provide acceptable accuracy in estimating multiple frequency response of the process from a step test because the number of input-output data is too small and the final part of the process excitation is not zero-steady-state.

Please, refer to the following my papers and book to estimate multiple frequency response models without any modeling errors theoretically. It can be applied to various process excitation only if the excitation signals belong to the specific types mentioned in the papers and book.

Sung, S. W., Lee, J., & Lee, I. B. (2009). Process Identifi cation and PID Control, Wiley-IEEE Press.

Cheon, Y. J., Jeon, C. H., Lee, J., Sung, S. W., & Lee, D. H. (2010). Improved Fourier transform for processes with initial cyclic-steady-state. AIChE Journal, 56(6), 1536-1544.

Cheon, Y. J., Sung, S. W., Lee, J., Je, C. H., & Lee, I. B. (2011). Improved frequency response model identi fication method for processes with initial cyclic-steady-state.

AIChE Journal, 57(12), 3429-3435.

The above-mentioned papers and book can be applied to continuous-time process. I expect that our lab could open other papers soon applicable to discrete-time processes. They are under revision.

Finally, we were able to find a solution to the problem. It involves finding an 'approximate'  DFT for the input and output signals. The key discovery was that the approximate DFT works better than an exact one.