Digital

Soma Biswas

2017

푃푎푟푡푖푎푙 푐푟푒푑푖푡 푓표푟 푠푙푖푑푒푠: 퐷푟. 푀푎푛표푗푖푡 푃푟푎푚푎푛푖푘 Changing the sampling rate

• A continuous-time signal 풙풄(풕) can be represented by a discrete-time signal consisting of a sequence of samples 풙 풏 = 풙풄(풏푻). • It is often necessary to change the sampling rate of a discrete-time signal, i.e., to obtain a new discrete-time representation of the underlying ′ ′ ′ continuous-time signal of the form 풙 풏 = 풙풄(풏푻 ), where 푻 ≠ 푻. • One approach is to obtain the sequences 풙′ 풏 from 풙 풏 is to reconstruct ′ ′ 풙풄(풕) from 풙 풏 and then resample 풙풄(풕) with period 푻 to obtain 풙 풏 . • However, this is not a desirable approach, because of the nonideal analog reconstruction filter, D/A converter, and A/D converter.

2 Sampling rate reduction by integer factor

• The sampling rate of a sequence can be reduced by “sampling” it. i.e., by defining a new sequence 풙풅 풏 = 풙 풏푴 = 풙풄(풏푴푻). • The system also called a sampling rate compressor or simply a compressor.

• If 푿풄 풋훀 = ퟎ 풇풐풓 훀 ≥ 훀푵, then 풙풅 풏 is an exact representation of 풙풄(풕) if 흅 흅 = ≥ 훀 . 푻′ 푴푻 푵 • That means the sampling rate can be reduced by a factor of 푴 without if the original sampling rate was at least 푴 times the or if the bandwidth of the sequence is first reduced by a factor of 푴 by discrete-time filtering. • In general, the operation of reducing the sampling rate (including any prefiltering) will be called downsampling.

3 푗휔 • 푋푑(푒 ) can be thought of as being composed of either an infinite set of copies of 푋푐(푗훺), 2휋 frequency scaled through 휔 = 훺푇′ and shifted by integer multiples of . 푇′ • Or 푀 copies of the periodic Fourier transform 푋(푒푗휔), frequency scaled by 푀 and shifted by integer multiples of 2휋. 4 5 • In general, to avoid aliasing in downsampling by a factor of M 휋 requires that 휔 푀 < 휋 표푟휔 < . 푁 푁 푀

• Downsampling can be done without aliasing if we are willing to reduce the bandwidth of the signal 푥[푛] before downsampling.

• If 푥[푛] is filtered by an ideal lowpass 휋 filter with cutoff frequency , then the 푀 output 푥 [푛] can be downsampled with aliasing.

• Note that the sequence 푥 푑 푛 = 푥 [푛푀] no longer represents the original underlyinjg continuout-time signal 푥푐(푡).

6 • Such system is called decimator, and downsampling by lowpass filtering followed by compression has been termed as decimation.

7 Increasing the sampling rate by integer factor

• The sampling rate of a sequence can be increased “sampling” it. i.e., by defining a new sequence 풙풊 풏 = 풙 풏/푳 = 풙풄(풏푻/푳). • The system on the left is called a sampling rate expander or simply a expander.

8 푗휔 푗휔 • 푋푖(푒 ) can be obtained from 푋푒(푒 ) by correcting the amplitude scale from 1 to 1 푇 푇′ and by removing all the frequency-scaled images of 푋푐(푗Ω) except at integer multiples of 2휋. • The system is therefore called interpolator, since it fills in the missing samples, and the operation of is therefore considered to be synonymous with interpolation.

9 Changing sampling rate by noninteger factor

• By combining decimation and interpolation, it is possible to change the sampling rate by a noninteger factor. • An interpolator that decreases the sampling period from 푇 to 푇/퐿, followed by a decimator that increases the sampling period by 푀, producing an output sequence 푥 푑 푛 that has an effective sampling period of 푇′ = 푇푀/퐿. • By choosing 퐿 and 푀 appropriately, one can get arbitrarily close to any desired ratio of sampling period. For example, if 퐿 = 100 and 푀 = 101, then 푇′ = 1.01푇.

10 11 Impulse invariance

• The impulse response of the discrete-time system is a scaled, sampled version of ℎ푐(푡). • When ℎ[푛] and ℎ푐(푡) are related like this, the discrete-time system is said to be an impulse-invariant version of the continuous-time system.

12 Example

13