[7.10 SUMMARY](#page-13-0)
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7.10 SUMMARY
In Ch. 6, we represented periodic signals as a sum of (everlasting) sinusoids or exponentials (Fourier series). In this chapter we extended this result to aperiodic signals, which are represented by the Fourier integral (instead of the Fourier series). An aperiodic signal x(t) may be regarded as a periodic signal with period T0 → ∞ so that the Fourier integral is basically a Fourier series with a fundamental frequency approaching zero. Therefore, for aperiodic signals, the Fourier spectra are continuous. This continuity means that a signal is represented as a sum of sinusoids (or exponentials) of all frequencies over a continuous frequency interval. The Fourier transform X(ω), therefore, is the spectral density (per unit bandwidth in hertz).
An ever-present aspect of the Fourier transform is the duality between time and frequency, which also implies duality between the signal x(t) and its transform X(ω). This duality arises because of near-symmetrical equations for direct and inverse Fourier transforms. The duality principle has far-reaching consequences and yields many valuable insights into signal analysis.
The scaling property of the Fourier transform leads to the conclusion that the signal bandwidth is inversely proportional to signal duration (signal width). Time shifting of a signal does not change its amplitude spectrum, but it does add a linear phase component to its spectrum. Multiplication of a signal by an exponential ejω0*t* shifts the spectrum to the right by ω0. In practice, spectral shifting is achieved by multiplying a signal by a sinusoid such as cosω0t (rather than the exponential ejω0*t* ). This process is known as amplitude modulation. Multiplication of two signals results in convolution of their spectra, whereas convolution of two signals results in multiplication of their spectra.
For an LTIC system with the frequency response H(ω), the input and output spectra X(ω) and Y(ω) are related by the equation Y(ω) = X(ω)H(ω). This is valid only for asymptotically stable systems. It also applies to marginally stable systems if the input does not contain a finite-amplitude sinusoid of the natural frequency of the system. For asymptotically unstable systems, the frequency response H(ω) does not exist. For distortionless transmission of a signal through an LTIC system, the amplitude response |H(ω)| of the system must be constant, and the phase response H(ω) should be a linear function of ω over a band of interest. Ideal filters, which allow distortionless transmission of a certain band of frequencies and suppress all the remaining frequencies, are physically unrealizable (noncausal). In fact, it is impossible to build a physical system with zero gain [H(ω) = 0] over a finite band of frequencies. Such systems (which include ideal filters) can be realized only with infinite time delay in the response.
The energy of a signal x(t) is equal to 1/2π times the area under |X(ω)2| (Parseval’s theorem). The energy contributed by spectral components within a band f (in hertz) is given by |X(ω)| 2f . Therefore, |X(ω)| 2 is the energy spectral density per unit bandwidth (in hertz).
The process of modulation shifts the signal spectrum to different frequencies. Modulation is used for many reasons: to transmit several messages simultaneously over the same channel for the sake of utilizing channel’s high bandwidth, to effectively radiate power over a radio link, to shift a signal spectrum at higher frequencies to overcome the difficulties associated with signal processing at lower frequencies, and to effect the exchange of transmission bandwidth and transmission power required to transmit data at a certain rate. Broadly speaking, there are two types of modulation, amplitude and angle modulation. Each class has several subclasses.
In practice, we often need to truncate data. Truncating is like viewing data through a window, which permits only certain portions of the data to be seen and hides (suppresses) the remainder. Abrupt truncation of data amounts to a rectangular window, which assigns a unit weight to data seen from the window and zero weight to the remaining data. Tapered windows, on the other hand, reduce the weight gradually from 1 to 0. Data truncation can cause some unsuspected problems. For example, in computation of the Fourier transform, windowing (data truncation) causes spectral spreading (spectral smearing) that is characteristic of the window function used. A rectangular window results in the least spreading, but it does so at the cost of a high and oscillatory spectral leakage outside the signal band, which decays slowly as 1/ω. In comparison to a rectangular window, tapered windows, in general, have larger spectral spreading (smearing), but the spectral leakage is smaller and decays faster with frequency. If we try to reduce spectral leakage by using a smoother window, the spectral spreading increases. Fortunately, spectral spreading can be reduced by increasing the window width. Therefore, we can achieve a given combination of spectral spread (transition bandwidth) and leakage characteristics by choosing a suitable tapered window function of a sufficiently long width T.