Difference between fourier series and z-transform?

Answer 1

Laplace = analogue signal

Fourier = digital signal

Notes on comparisons between Fourier and Laplace transforms:

The Laplace transform of a function is just like the Fourier transform of the same function, except for two things. The term in the exponential of a Laplace transform is a complex number instead of just an imaginary number and the lower limit of integration doesn't need to start at -∞. The exponential factor has the effect of forcing the signals to converge. That is why the Laplace transform can be applied to a broader class of signals than the Fourier transform, including exponentially growing signals. In a Fourier transform, both the signal in time domain and its spectrum in frequency domain are a one-dimensional, complex function. However, the Laplace transform of the 1D signal is a complex function defined over a two-dimensional complex plane, called the s-plane, spanned by two variables, one for the horizontal real axis and one for the vertical imaginary axis. If this 2D function is evaluated along the imaginary axis, the Laplace transform simply becomes the Fourier transform.

Answer 2

The Fourier series is an expression of a pattern (such as an electrical waveform or signal) in terms of a group of sine or cosine waves of different frequencies and amplitude. This is the frequency domain.

The Fourier transform is the process or function used to convert from time domain (example: voltage samples over time, as you see on an oscilloscope) to the frequency domain, which you see on a graphic equalizer or spectrum analyzer.

The inverse Fourier transform converts the frequency domain results back to time domain. The use of transforms is not limited to voltages.

Answer 3

A Fourier series is a series of sine and cosine harmonics of a particular frequency.

For example sinf+icosf + 3 sin2f+ 5icos2f... where the successive terms are multiples of the fundamental frequency f. It is typical ( but as far as I know not required) that complex numbers are used.

A Fourier transform converts a time domain wave form (like a sound wave) into the coefficients of the corresponding Fourier series.

A DFT is a digital approximation to a Fourier transform, usually using something like the Cooley-Tuckey Fast Fourier Transform (FFT) for efficiency.

The underlying Fourier theorem is something like: Every bounded periodic continuous (needed to avoid Gibbs) function , or wave form, can be written as the sum of its Fourier series. i.e. It is a sum of sines and cosines

In otherwords, you take a wave form in the time domain like a sound wave and break it into its components (various frequencies) by the Fourier Transform. The results of the Transform are the coefficients of the Fourier series. The wave form of a voice converted to components (and perhaps a little more) is a voiceprint.

Answer 4

fourier transform deals with continuous time non-periodic signals(if the input is continuous time periodic signal(time domain) then the output is continuous nonpeiodic signal (frequency domain))

DTFT-input-discrete time nonperoidic signal(time domain),output-continuous peroidic signal(frequency domain)

Answer 5

As it has been already hinted, Fourier Series is used for periodic signals. It represents the signal by the discrete-time sequence of basis functions with finite and concrete amplitude and phase shift. The basis functions, according to the theory, are harmonics with the frequencies, divisible by the frequency of the signal (which coincides with the frequency of the 1st main harmonic). All the harmonics with the number>1 are called higher harmonics, whereas the 1st one is called - the main harmonic. After reminding the mathematical properties of the signal we can maintain, that sometimes harmonics with even or odd numbers are absent at all. There phases are sometimes always equal to 0 and 180 degrees or to 90 and -90 degrees.

Fourier series are known to exist in sinus-cosinus form, sinus form, cosinus form, complex form. The choice depends on the problem solved and must be convenient for further analysis.

Fourier tranform is invented and adjusted for aperiodic signals with integrated absolute value and satisfaction of Diricle conditions. It's worth saying, that Dirichle conditions is the necessary requirement for Fourier series too. Fourier representation of aperiodic signals is not discrete, but continious and the amplitudes are infinitely small. They play the role of the proportional coefficients.

there are links between Fourier series of periodic signal and Fourier transform. These links may be easily found in almost all the books on classical Fourier analysis of signals. For example, see Oppenheim, Djervis and others.

Answer 6

z transform is related to discrete time signal while fourier series is related to continuous time signal.

z transform=sigmalm -infinty to +infinity x(n)z-n