An ideal low-pass filter (LPF) is specified by in .NET framework

Drawer Quick Response Code in .NET framework An ideal low-pass filter (LPF) is specified by

An ideal low-pass filter (LPF) is specified by
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which is shown in Fig. 5-5(a). The frequency o, is called the cutoff frequency.
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2. Ideal High-Pass Filter:
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An ideal high-pass filter (HPF) is specified by
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which is shown in Fig. 5-5(b).
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FOURIER ANALYSIS OF TIME SIGNALS AND SYSTEMS
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Fig. 5-5
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Magnitude responses of ideal frequency-selective filters.
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3. Ideal Bandpass Filter:
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An ideal bandpass filter (BPF) is specified by
IH(w)l=
< lwl < 0
otherwise
which is shown in Fig. 5-5(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. 5-5(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 frequency-selective filters are noncausal systems.
CHAP. 51
FOURIER ANALYSIS OF TIME SIGNALS AND SYSTEMS
B. Nonideal Frequency-Selective Filters:
As an example of a simple continuous-time causal frequency-selective filter, we consider the RC filter shown in Fig. 5-6(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. 5-6 RC filter and its frequency response.
FOURIER ANALYSIS OF TIME SIGNALS AND SYSTEMS
[CHAP. 5
which are plotted in Fig. 5-6(b). From Fig. 5-6(b) we see that the RC network in Fig. 5-6(a) performs as a low-pass 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 low-pass filter equals its cutoff frequency; that is, WB= w, [Fig. 5-5(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. 5-5(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 high-pass or a bandstop filter.
2. 3-dB (or Half-Power) Bandwidth:
For causal or practical filters, a common definition of filter (or system) bandwidth is In the case of a low-pass filter, such as the RC filter described the 3-dB bandwidth W , is defined as the positive frequency at which by Eq. (5.92) or in Fig. 5-6(b), W, the amplitude spectrum IH(w)l drops to a value equal to I H ( o ) I / ~ ,as illustrated in Fig. 5-7(a). Note that (H(O)I is the peak value of H ( o ) for the low-pass RC filter. The 3-dB bandwidth is also known as the half-power 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. 5-7(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 one-sided, bandwidth only.
Fig. 5-7 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).
3-dB Bandwidth:
The bandwidth of a signal x ( t ) can also be defined on a similar basis as a filter bandwidth such as the 3-dB bandwidth, using the magnitude spectrum (X(o)lof the signal. Indeed, if we replace IH(o)l by IX(o)l in Figs. 5-5(a) to ( c ) , we have frequency-domain plots of low-pass, high-pass, and bandpass signals.
Band-Limited Signal:
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