Fundamentals of Digital Signal Processing

Lecture 10 Sampling Theorem 3

Fundamentals of Digital Signal ProcessingSpring, 2012

Wei-Ta Chu2012/4/10

1 DSP, CSIE, CCU

Interpolation with Pulses� Obviously, the important issue is the choice of the

pulse waveform p(t).

� Four possible pulse waveformsfor D-to-C conversion whenfs = 200 Hz

� T = 5 msec

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� Ts = 5 msec

Zero-Order Hold Interpolation� The simplest pulse shape is a symmetric square pulse

� In this example, the total widthof the square pulse is Ts=5 msand its amplitude is 1.

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and its amplitude is 1.

� Since a constant is a polynomial of zero order, and since the effect of the flat pulse is to hold or replace each sample for Ts sec, then used of a flat pulse is called a zero-order hold reconstruction

Linear Interpolation� The triangular pulse defined as a pulse consisting of

the first-order polynomial segments

� The duration of the pulse is 2Ts, and they are shifted by multiples

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and they are shifted by multiples of Ts.

� The result is that the samples areconnected by straight lines.

Cubic SplineInterpolation� The cubic spline pulse has a duration twice of the triangular

pulse and four times of the square pulse. � This pulse has zeros at the key

locations:

� For values of t, four pulses overlap and must be added

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overlap and must be added together

� The reconstructed signal at a particular time instant is the sumof these overlapping pulses, depending on two proceeding samples and two following samples

Over-Sampling Aids Interpolation� One way to make a smooth reconstruction is to use p(t)

that is smooth and has a long duration.

� If the original waveform does not vary much over the duration of p(t), then we will also obtain a good reconstruction → oversampling

� Change the sampling frequency from f =200 Hz to 500

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� Change the sampling frequency from fs=200 Hz to 500 Hz in the previous examples

� The duration of reconstruction pulses is changed from Ts=5 msec to 2 msec

Over-Sampling Aids Interpolation

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Ideal BandlimitedInterpolation� What is the pulse shape that gives “ideal D-to-C

conversion”? (Chapter 12)

� The infinite length of this pulse implies that to reconstruct a signal at time t exactly from its samples

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reconstruct a signal at time t exactly from its samples requires all the samples, not just those around that time.

� It’s called bandlimited interpolation

� Using this pulse to reconstruct from samples of a cosine wave will always produce a cosine wave exactly.

The Sampling Theorem

A continuous-time signalx(t) with frequenciesno higher thanfmax can be reconstructed exactlyfrom its samplesx[n]=x(nTs), if the samples aretaken at a ratefs=1/Ts that is greater than 2fmax.

Shannon Sampling Theorem

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The Sampling Theorem� If the input is composed of sinusoidal signals limited

to the set of frequencies in the range , then the reconstructed signal is equal to the original signal that was sampled; i.e., y(t) = x(t).

� Signals composed of sinusoids such that all frequencies are limited to a “band of frequencies” are called bandlimited signals.

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called bandlimited signals.

� Such an additive combination of cosine signals can produce an infinite variety of both periodic and nonperiodic signal waveforms.

The Sampling Theorem� If we sample the signal

� If we sample a sum of continuous-time cosines, we obtain a sum of sampled cosines each of which would

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obtain a sum of sampled cosines each of which would be subject to aliasing if the sampling rate is not high enough

� Discrete-to-continuous conversion by interpolation

The Sampling Theorem

� Since each individual sinusoid is assumed to satisfy the conditions of the sampling theorem, it follows that the D-

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to-C converter will reconstruct each component perfectly

� The Shannon sampling theorem applies to any signal that can be represented as a bandlimited sum of sinusoids.

Homework 3� Chapter 4: P-4.1, 4.8, 4.13, 4.14

� Hand over your homework in the class at Apr. 17

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Fundamentals of Digital Signal Processing

Lecture 10 Introduction of FIR Filters

Fundamentals of Digital Signal ProcessingSpring, 2012

Wei-Ta Chu2012/4/10

14 DSP, CSIE, CCU

Discrete-Time System� A discrete-time system is a computational process for

transforming one sequence, called the input signal, into another sequence called the output signal.

� The output sequence is related to the input sequence by a process that can be described mathematically by an

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a process that can be described mathematically by an operator

� Examples:

FIR Filter� A filter is a system that is designed to remove some

component or modify some characteristic of a signal, but often two terms are used interchangeably.

� FIR (finite impulse response) systems are systems for which each output sample is the sum of a finite number of weighted samples of the input sequence.

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number of weighted samples of the input sequence.

The Running-Average Filter� A simple FIR filter is to compute a moving average or

running average of two or more consecutive numbers of the sequence, thereby forming a new sequence of the average values.

� Example: the input is a finite-length signal, the supportof such a sequence is the set of values over which the

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of such a sequence is the set of values over which the sequence is nonzero. � Support: the finite interval

The Running-Average Filter� Output:

� The equation is called a difference equation. It’s a complete description of the FIR system because we can use it to

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description of the FIR system because we can use it to compute the entire output signal for

n n<-2 -2 -1 0 1 2 3 4 5 n>5

x[n] 0 0 0 2 4 6 4 2 0 0

y[n] 0 2/3 2 4 14/3 4 2 2/3 0 0

The Running-Average Filter� y[n] = 0 outside of the finite interval , i.e.

the output also has finite support� The output sequence is longer than the input sequence,

and that the output appears to be a somewhat rounded-off version of the input; i.e. it is smoother than the input sequence. This behavior is characteristic of running-average FIR filter.

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running-average FIR filter. � The output starts (becomes nonzero) before the input

starts. � n would stand for time� Present output: y[n]� Inputs are indexed as n, n+1, n+2. Two of them are “in

the future”.

The Running-Average Filter� In general, values from either the pastor the futureor

both may be used in the computation.

� A filter that uses only the present and past values of the input is called a causal filter.

� A filter that uses future values of the input is called noncausal.

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noncausal.

The Running-Average Filter� Change the difference equation

� It’s a causal running averager, or it may well be called a backward average.

n n<-2 -2 -1 0 1 2 3 4 5 6 7 n>7

x[n] 0 0 0 2 4 6 4 2 0 0 0 0

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� The output of the causal filter is simply a shifted version of the output of the previous noncausal filter.

� The output doesn’t change from zero before the input changes from zero

x[n] 0 0 0 2 4 6 4 2 0 0 0 0

y[n] 0 0 0 2/3 2 4 14/3 4 2 2/3 0 0

The General FIR Filter� If

� Then we have a length-4 filter with M = 3

� The parameter M is the order of the FIR filter. The

Mth order FIR filter

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� The parameter M is the order of the FIR filter. The number of filter coefficients is also called the filter length (L). The length is one greater than the order, i.e. L=M+1

An Illustration of FIR Filtering� Consider an input signal

� We often have real signals of this form: a component that is of interest (it may be the slowly varying exponential component (1.02)n) plus another

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exponential component (1.02)n) plus another component that is not of interest. � The second component is often considered to be noise

that interferes with observation of the desired signal.

An Illustration of FIR Filtering� Suppose that x[n] is the input to a

causal 3-point running averager

� (a)the output must be zero for n<0

� (b)shaded interval of length M=2 samples at the beginning

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� (c) shaded interval of length M=2 samples at the end

� (d)the solid line showing the values of the exponential component has been shifted to the right by M/2=1 sample

An Illustration of FIR Filtering� 7-point running averager

� (a) shaded region at the beginning and the end

� (b) the exponential component is very close to

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component is very close to the exponential component of the input (after a shift of M/2=3 samples)

Summary� (1) FIR filtering can modify signals in ways that may

be useful

� (2) The length of the averaging interval seems to have a big effect on the resulting output

� (3) The running-average filters appear to introduce a shift equal to M/2 samples

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shift equal to M/2 samples

� We will gain full appreciation of this example only upon the completion of Chapter 6.

The Unit Impulse Sequence� The unit impulse is perhaps the simplest sequence

because it has only one nonzero values, which occurs at n = 0. The mathematical notation is that of the Kronecker delta function

n … -2 -1 0 1 2 3 4 …

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n … -2 -1 0 1 2 3 4 …

0 0 0 1 0 0 0 0 0

0 0 0 0 0 1 0 0 0

The Unit Impulse Sequence� The shifted impulse is a concept that is very useful in

representing signals and systems.

n … -2 -1 0 1 2 3 4 …

0 0 0 2 0 0 0 0 0

0 0 0 0 4 0 0 0 0

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0 0 0 0 4 0 0 0 0

0 0 0 0 0 6 0 0 0

0 0 0 0 0 0 4 0 0

0 0 0 0 0 0 0 2 0

0 0 0 2 4 6 4 2 0

The Unit Impulse Sequence� Any sequence can be represented in this way. The

equation

is true if k ranges over all the nonzero values of the sequence x[n].

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sequence x[n].

� The sequence is formed by using scaled shifted impulses to place samples of the right size at the right positions.

Unit Impulse Response Sequence� When the input to the FIR filter is a unit impulse

sequence, , the output is the unit impulse response, denoted as h[n].

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n <0 0 1 2 3 … M M+1 …

0 1 0 0 0 0 0 0 0

0 b0 b1 b2 b3 … bM 0 0

Unit Impulse Response Sequence� The impulse response h[n] of the FIR filter is simply the

sequence of difference equation coefficients.

� Since h[n] = 0 for n<0 and for n>M, the length of the impulse response sequence h[n] is finite. This is why the system is called a finite impulse response (FIR) system.

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The plot of the impulse response for the case of the causal 3-point running-average filter.