Capitulo 3 - Señales

Diapositiva 1

UNIVERSIDAD POLITECNICA SALESIANA

Semestre: Septiembre Febrero 2010Ing. Walter Orozco. [email protected]/05/20101Seales y Sistemas - Ing. Walter Orozco. Msc.ANLISIS DE LA TRANSFORMADA DE FOURIER DE SEALES Y SISTEMAS

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CAPITULO 3Anlisis de la Transformada de Fourier de Seales y Sistemas3.1Introduccion3.2 Respuesta en frecuencia3.3 Filtros ideales3.4 Filtros Pasivos prcticos3.5 Grafica de magnitud logartmica de la respuesta en frecuencia y diagramas de Bode3.6 Filtros prcticos activos3.7 Anlisis espectral3.8 Ejercicios

13/05/2010Seales y Sistemas - Ing. Walter Orozco. Msc.2


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13/05/2010Seales y Sistemas - Ing. Walter Orozco. Msc.33.1IntroductionIn this chapter we begin the discussion by considering the issue of distortionless transmission, which is basic to the study of linear filters and equalizers. This lead naturally to a discussion of an idealized framework for filtering, which, in turn, provides the basis for the design of practical filters.

The design of a filter can be accomplished by using continuous-time concepts, in which case we speak of analog filters. Alternatively, the design can be accomplished by using discrete-time concepts, in which case we speak of digital filter. 3.2 Frequency Analysis

3.2.1Conditions for distortionless transmissionConsider a continuous-time LTI system:

Frequency response

Input signal

Output signal


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13/05/2010Seales y Sistemas - Ing. Walter Orozco. Msc.4We wish to know conditions for distortionless transmission through the system. By distortionless transmission we mean that the output signal of the system is an exactly replica of the input signal, except, possibly, for two minor modifications.

A scaling of amplitudeA constant time delayWe say that a signal x(t) is transmitted through the system without distortion if the output signal y(t) is defined by:

Where the constant C accounts for a change in amplitude and the constant t0 accounts for a delay in transmission.

Fourier Transform


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13/05/2010Seales y Sistemas - Ing. Walter Orozco. Msc.5The frequency response of a distortionless LTI system is therefore:

Correspondingly, the impulse response of the system is given by:

The Last two equations describe the frequency-domain and time-domain conditions, respectively, that an LTI system has to satisfy for distortionless transmission.

Form a practical viewpoint, the frequency-domain equation is the more reveling of the two, indicating that, in order to achieve the distortionless transmission of a signal with some finite frequency content through a continuous-time LTI system, the frequency response of the system must satisfy two conditions.


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13/05/2010Seales y Sistemas - Ing. Walter Orozco. Msc.61. The magnitude response must be a constant for all frequencies of interest, that is, we must have:

2. For the same frequencies of interest, the phase response must be linear in frequency, with slope t0 and intercept zero; that is, we must have:

These two conditions are illustrated in the figures


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13/05/2010Seales y Sistemas - Ing. Walter Orozco. Msc.73.3 Ideal FiltersThe frequency response of a filter is characterized by a pass band and a stop band, which are separated by a transition band, also known a guard band. Signals with frequencies inside the pass band are transmitted with little or not distortion, whereas those with frequencies inside the stop band are effectively rejected.

The filter may thus be of the:

a)Los-pass (Transmits low frequencies)b)High pass (Transmits high frequencies)c)Band-pass (Transmits intermediate frequencies)d)Band-stop (Transmits all but intermediate frequencies)


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The frequency response of an ideal low-pass filter wit cutoff frequency C is defined by:


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13/05/2010Seales y Sistemas - Ing. Walter Orozco. Msc.9To evaluate h(t), we take the inverse Fourier transform of the last equation:

We may rewrite the equation in the compact form

For a finite delay t0, the ideal low-pass filter is non causal.


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Impulse responses of different type of filters


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Causal Band pass

Causal Band stop

Causal High Pass


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13/05/2010Seales y Sistemas - Ing. Walter Orozco. Msc.123.3.1 Design of filtersThe low-pass filter with frequency response shown in the last slides is ideal in that it passes all frequency components lying inside the pass band with no distortion and rejects all frequency components lying inside the stop band, and the transition from the pass band to the stop band is abrupt.

Therefore, form a practical perspective, the prudent approach is to tolerate an acceptable level of distortion by permitting prescribed deviations from these ideal conditions, as described here for the case of continuous-time o analog filters:

1. Inside the pass band, the magnitude response of the filter should lie between 1 and 1- that is:

Where p is the pass band cutoff frequency and is tolerance parameter.

2. Inside the stop band, the magnitude response of the filter should no exceed ; that is

Where s is the stop band cutoff frequency and is another tolerance parameter.

3. The transition bandwidth has a finite width to s - p


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13/05/2010Seales y Sistemas - Ing. Walter Orozco. Msc.13Tolerance diagram of a practical low-pass filter: The pass band; transition band , and stop band are shown for positive frequencies.Having formulated a set of specifications describing the desired properties of the frequency-selective filter, we set two distinct steps involved in the design of the filter, pursued in the following order:

The approximation of a prescribed frequency response by a rational transfer function which represents a system that is both causal and stable.The realization of the approximating transfer function by a physical system.


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13/05/2010Seales y Sistemas - Ing. Walter Orozco. Msc.14There are three different approaches to the design of a analog filter, as summarized here:

Analog approach, which applies to the class of analog filters.Analog -to-digital approach, where the motivation is to design a filter by building on what we know about an analog filter design.Direct digital approach, which applies to the class of digital filters.


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13/05/2010Seales y Sistemas - Ing. Walter Orozco. Msc.153.4 Filtros pasivos prcticos.Filtro pasa bajo RCAproximacin del filtro ideal mediante circuitos electrnicos.

Tomando la transformada de Fourier


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13/05/2010Seales y Sistemas - Ing. Walter Orozco. Msc.16Aplicando la transformada de Fourier a la relacin de voltajes y corrientes.

Se expresa el circuito en funcin exclusivamente de impedancias.


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Grafica de magnitud y fase de un filtro pasa bajo implementado con circuitos electrnicos.

Tomando la transformada inversa de Fourier

Respuesta al impulso del filtro


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Respuesta al impulso del filtro

Operacin fsica del filtro pasa bajo:A frecuencias muy bajas:

A frecuencias muy altas:


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13/05/2010Seales y Sistemas - Ing. Walter Orozco. Msc.19Anlisis del filtro pasa bajo cuya seal de entrada es voltaje y cuya seal de salida es la corriente del circuito.

Operacin fsica del filtro pasa bajo:A frecuencias muy bajas:


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13/05/2010Seales y Sistemas - Ing. Walter Orozco. Msc.20Operacin fsica del filtro pasa bajo:A frecuencias muy altas:

Comparando el comportamiento del filtro en el contexto voltaje voltaje, con respecto al corriente voltaje, se observa que el comportamiento del filtro ahora es de un pasa alto.

A altas frecuencias el flujo de corriente se controla por medio de la Resistencia.

Forma alternativa de un filtro pasa bajo.


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13/05/2010Seales y Sistemas - Ing. Walter Orozco. Msc.213.5 Grafica de magnitud logartmica de la respuesta en frecuencia y diagramas de BodeEn muchas ocasiones las graficas lineales de la respuesta en frecuencia, aunque son exactas, no revelan aspectos importantes en el comportamiento de los sistemas, los cuales no son percibidos, por tanto al usar escalas logartmicas es posible observar aspectos relevantes en un sistema. Para tal efecto se utilizan los diagramas o graficas de bode.

Diagrama de bloques de la funcin de transferencia factorizada.


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13/05/2010Seales y Sistemas - Ing. Walter Orozco. Msc.22Otras de las ventajas que se tienen al utilizar logaritmos es que el producto y divisin de factores tan solo se reduce a sumas y restas, de tal forma que la funcin de transferencia queda convertida a:

MagnitudFase


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13/05/2010Seales y Sistemas - Ing. Walter Orozco. Msc.23Para el anlisis del diagrama de bloques consideraremos la siguiente funcin de transferencia.

Parte I:Parte II:Sistemas de un polo real


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Parte I:Parte II:Sistemas de un cero real


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IntegradorDiferenciador


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Ganancia independiente de la frecuencia.


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Pares de polos complejos


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Pares de ceros complejos


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13/05/2010Seales y Sistemas - Ing. Walter Orozco. Msc.29Ejemplo: Grafique el diagrama de Bode para la funcin de transferencia de voltaje del circuito, donde C1=1 F, C2=2 F, Rs=4 , R1=2 , R2=3


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13/05/2010Seales y Sistemas - Ing. Walter Orozco. Msc.303.6 Filtros prcticos activos

UNIVERSIDAD POLITECNICA SALESIANAHay dos formas de circuitos amplificador operacional que se usan comnmente:La de amplificador inversor.La de no inversor.

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13/05/2010Seales y Sistemas - Ing. Walter Orozco. Msc.31Todos los filtros prcticos que se han examinado hasta ahora han sido pasivos. Pasivos significa que no contienen dispositivos con la capacidad de tener una respuesta con ms potencia real que la excitacin.

Muchos filtros modernos son activos. Esto es, contienen dispositivos activos como transistores y amplificadores operacionales y requieren una fuente externa de potencia para operar en forma apropiada. Con el uso de dispositivos activos la potencia de la respuesta real puede ser mayor que la potencia de la excitacin real.

Integracin del voltaje de salidaIntegrador activo


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13/05/2010Seales y Sistemas - Ing. Walter Orozco. Msc.32El integrador se transforma fcilmente en un filtro pasa bajo mediante la adicin de un solo resistor.


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UNIVERSIDAD POLITECNICA SALESIANAEjercicio para realizarlo en clase:Determinar la funcin de transferencia del siguiente filtro y grafique su diagrama de Bode.

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13/05/2010Seales y Sistemas - Ing. Walter Orozco. Msc.343.7 Anlisis espectral (Trabajo de investigacin)

UNIVERSIDAD POLITECNICA SALESIANAEn el sistema de la figura, , y la frecuencia del filtro pasa bajas es 1Hz.

Grafique las seales:(a)(b) (c)(d)

Adems grafique las magnitudes y fases de las seales indicadas en los literales (a), (b), (c) y (d).

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13/05/2010Seales y Sistemas - Ing. Walter Orozco. Msc.35Bibliografa[1] Roberts M.J. Seales y Sistemas, McGraw-Hill Interamericana, Primera Edicin, 2005.

[2] Haykin S, Van Veen B. Signal and Systems, Jhon Wiley and Sons, Inc. Second Edition, 2003.


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Capitulo 3 - Señales