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An ideal lowpass filter (LPF) is specified by in .NET framework
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IH(w)l=
< lwl < 0
otherwise
which is shown in Fig. 55(c). 4. Ideal Bandstop Filter: An ideal bandstop filter (BSF) is specified by
H(w)1= < 11< w 2 0
otherwise
which is shown in Fig. 55(d). In the above discussion, we said nothing regarding the phase response of the filters. T o avoid phase distortion in the filtering process, a filter should have a linear phase characteristic over the pass band of the filter, that is [Eq. (5.82b11, where t , is a constant. Note that all ideal frequencyselective filters are noncausal systems.
CHAP. 51
FOURIER ANALYSIS OF TIME SIGNALS AND SYSTEMS
B. Nonideal FrequencySelective Filters: As an example of a simple continuoustime causal frequencyselective filter, we consider the RC filter shown in Fig. 56(a).The output y(t) and the input x ( t ) are related by (Prob. 1.32) Taking the Fourier transforms of both sides of the above equation, the frequency response H(w) of the RC filter is given by where w 0 = 1/RC. Thus, the amplitude response (H(w)l and phase response OJw) are given by
Fig. 56 RC filter and its frequency response.
FOURIER ANALYSIS OF TIME SIGNALS AND SYSTEMS
[CHAP. 5
which are plotted in Fig. 56(b). From Fig. 56(b) we see that the RC network in Fig. 56(a) performs as a lowpass filter. 5.7 BANDWIDTH
A. Filter (or System) Bandwidth: One important concept in system analysis is the bandwidth of an LTI system. There are many different definitions of system bandwidth. I . Absolute Bandwidth: The bandwidth WB of an ideal lowpass filter equals its cutoff frequency; that is, WB= w, [Fig. 55(a)]. In this case W, is called the absolute bandwidth. The absolute bandwidth of an ideal bandpass filter is given by W, = w 2  w , [Fig. 55(c)]. A bandpass < ( filter is called narrowband if W, < w,, where w,, = ; w , + w 2 ) is the center frequency of the filter. No bandwidth is defined for a highpass or a bandstop filter. 2. 3dB (or HalfPower) Bandwidth: For causal or practical filters, a common definition of filter (or system) bandwidth is In the case of a lowpass filter, such as the RC filter described the 3dB bandwidth W , is defined as the positive frequency at which by Eq. (5.92) or in Fig. 56(b), W, the amplitude spectrum IH(w)l drops to a value equal to I H ( o ) I / ~ ,as illustrated in Fig. 57(a). Note that (H(O)I is the peak value of H ( o ) for the lowpass RC filter. The 3dB bandwidth is also known as the halfpower bandwidth because a voltage or current attenuation of 3 dB is equivalent to a power attenuation by a factor of 2. In the case of a bandpass filter, W, is defined as the difference between the frequencies at which )H(w)l / drops to a value equal to 1 a times the peak value IH(w,)l as illustrated in Fig. 57(b). This definition of W , is useful for systems with unimodal amplitude response (in the positive frequency range) and is a widely accepted criterion for measuring a system's bandwidth, but it may become ambiguous and nonunique with systems having multiple peak amplitude responses. Note that each of the preceding bandwidth definitions is defined along the positive frequency axis only and always defines positive frequency, or onesided, bandwidth only. Fig. 57 Filter bandwidth.
CHAP. 51
FOURIER ANALYSIS O F TIME SIGNALS AND SYSTEMS
B. Signal Bandwidth: The bandwidth of a signal can be defined as the range of positive frequencies in which "most" of the energy or power lies. This definition is rather ambiguous and is subject to various conventions (Probs. 5.57 and 5.76). 3dB Bandwidth: The bandwidth of a signal x ( t ) can also be defined on a similar basis as a filter bandwidth such as the 3dB bandwidth, using the magnitude spectrum (X(o)lof the signal. Indeed, if we replace IH(o)l by IX(o)l in Figs. 55(a) to ( c ) , we have frequencydomain plots of lowpass, highpass, and bandpass signals. BandLimited Signal:

