Properties of Fourier representation
The main properties of Fourier representation are as follows:
The Convolution Property
One of the most important properties of Fourier representations is the convolution property. The convolution of signals in the time domain transforms to the multiplication of their respective Fourier representations in the frequency domain. With the convolution property, we can analyze the input-output behavior of a linear system in the frequency domain by multiplying transforms instead of convolving time signals.
This can significantly simplify system analysis and offers considerable insight into the system behavior. It is a consequence of complex sinusoids being eigenfunctions of LTI systems. One of the best ways to start is to examine the convolution property as applied to nonperiodic signals, which is discussed in later articles.
Filtering
The multiplication that takes place in the frequency-domain representation gives rise to the notion of filtering. A system presents a different response to components of the input that are at different frequencies in order to perform filtering. Typically, the term “filtering” implies that some frequency components of the input are eliminated while others are passed by the system unchanged. The systems can be described in terms of the type of filtering that they perform on the input signal. A low-pass filter attenuates the high-frequency components of the input and passes the lower frequency components.
In contrast, a high pass filter attenuates low frequencies and passes the high frequencies. A band-pass filter passes signals within a certain frequency band and attenuates signals outside that band. The characterization of the discrete-time filter is based on its behavior in the frequency range -pi< Ω <= pi because its frequency response is periodic. Hence a high-pass discrete-time filter passes frequencies near w and attenuates frequencies near zero.
The passband of a filter is the band of frequencies that are passed by the system, while the stopband refers to the range of frequencies that are attenuated by the system. It is impossible to build a practical system that has discontinuous frequency response characteristics.
Realistic filters always have a gradual transition from the passband to the stopband. The range of frequencies over which this occurs is known as the transition band. Furthermore, realistic filters do not have zero gain over the entire stopband, but instead, have a very small gain relative to that of the passband. In general, filters with sharp transitions from passband to stopband are more difficult to implement.
The magnitude response of a filter is commonly described in units of decibels, or dB.
The magnitude response in the stopband is normally much smaller than that in the passband, and the details of the stopband response are difficult to visualize on a linear scale. By using decibels, we display the magnitude response on a logarithmic scale and are able to examine the details of the response in both the passband and the stopband.
It can be noted that a unity gain corresponds to zero dB. Therefore, the magnitude response in the filter passband is normally close to zero dB. The edge of the passband is usually defined by the frequencies for which the response is -3 dB, corresponding to a magnitude response of (1/sqrt(2)) the energy spectrum of the filter. As the output is given by:
|Y(jw)| ^2 = |H(jw)|^2 |X(jw)|^2
The -3-dB point corresponds to the frequency at which the filter passes only half of the input power. The -3-dB points have usually termed the cut-off frequencies of the filter. The majority of filtering applications involve real-valued impulse responses, which implies magnitude responses with even symmetry.
Differentiation and integration properties
Differentiation and integration are operations that apply to continuous functions. Hence, we may consider the effect of differentiation and integration with respect to time for a continuous-time signal or with respect to frequency in the FT and DTFT, since these representations are continuous functions of frequency.
Frequency-shift properties
A frequency shift corresponds to multiplication in the time domain by a complex sinusoid whose frequency is equal to the shift. This property is a consequence of the frequency-shift properties of the complex sinusoid. A shift in the frequency of a complex sinusoid is equivalent to a multiplication of the original complex sinusoid by another complex sinusoid whose frequency is equal to the shift.
Since all the Fourier representations are based on complex sinusoids. The frequency shift must be integer-valued in both Fourier series cases. This leads to multiplication by a complex sinusoid whose frequency is an integer multiple of the fundamental frequency. The other observation is that the frequency-shift property is the “dual” of the time-shift property. We may summarize both properties by stating that a shift in one domain, either frequency or time, leads to a multiplication by a complex sinusoid in the other domain.
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