Frequency Response of FIR FiltersSinusoidal Response of FIR SystemsSuperposition and the Frequency ResponseSteady-State and Transient ResponseProperties of the Frequency ResponseRelation to Impulse Response and Difference EquationPeriodicity ofConjugate SymmetryGraphical Representation of the Frequency ResponseCascaded LTI SystemsMoving Average FilteringPlotting the Frequency ResponseFiltering Sampled Continuous-Time SignalsInterpretation of DelayECE 2610 Signal and Systems 6–1Frequency Response of FIR FiltersThis chapter continues the study of FIR filters from Chapter 5,but the emphasis is frequency response, which relates to how thefilter responds to an input of the form.A fundamental result we shall soon see, is that the frequencyresponse and impulse response are related through an operationknown as the Fourier transform. The Fourier transform itself isnot however formally studied in this chapter.Sinusoidal Response of FIR Systems• Consider an FIR filter when the input is a complex sinusoidof the form, (6.1)where it could be that was obtained by sampling thecomplex sinusoid and • From the difference equation for an tap FIR filter,(6.2)xn ejˆ0n– n =xn Aejejˆ0n– n =xnxt Aejej0t=ˆ00Ts=M 1+yn bkxn k–k 0=MbkAejejˆ0nk–k 0=M==Chapter6Sinusoidal Response of FIR SystemsECE 2610 Signals and Systems 6–2(6.3)where we have defined for arbitrary (6.4)to be the frequency response of the FIR filter• Note: The notation is used rather than say , tobe consistent with the z-transform which will defined inChapter 7, and to emphasize the fact that the frequencyresponse is periodic, with period (more on this later)– Note: , where sine and cosine areboth functions• Returning to (6.2), the implication is that when the input is acomplex exponential at frequency , the output is also acomplex exponential at frequency • The complex amplitude (magnitude and phase) of the input ischanged as a result of passing through the system– The frequency response at multiplies the input ampli-tude to produce the output– It is not true in general that , butonly for the special case of a complex sinusoidapplied starting at yn Aejejˆ0nbkej– ˆ0kk 0=M=Aejejˆ0nHejˆ0– n =ˆHejˆ bkej– ˆkk 0=M=HejˆH ˆ2ejˆˆ j ˆsin+cos=mod 2ˆ0ˆ0ˆ0yn Hejˆ0xn=xn–Sinusoidal Response of FIR SystemsECE 2610 Signals and Systems 6–3• The frequency response is a complex function of that isgenerally viewed in either polar or rectangular form(6.5)– The polar or magnitude and phase form is perhaps themost common• The polar form offers the following interpretation of interms of , when the input is a complex sinusoid(6.6)– Here we see that the input amplitude is multiplied by thefrequency response amplitude, and the input phase hasadded to it the frequency response phase– The output amplitude expression means that isalso termed the gain of an LTI systemExample: • The frequency response of this FIR filter isˆHejˆ HejˆejHejˆ=ReHejˆjImHejˆ+=ynxnyn HejˆejHejˆAejejˆn=HejˆAejHejˆ+ejˆn=Hejˆbk 11311=Hejˆ bkejˆk–k 0=4=1 ejˆ–3ej– 2ˆej– 3ˆej– 4ˆ++ + +=Sinusoidal Response of FIR SystemsECE 2610 Signals and Systems 6–4– We have used the inverse Euler formula for cosine twice• For this particular filter we have thatWhy?•Use MATLAB to plot the magnitude and phase response>> w = 0:2*pi/200:2*pi;>> H = exp(-j*2*w).*(3 + 2*cos(w) + 2*cos(2*w));>> subplot(211)>> plot(w,abs(H))>> axis([0 2*pi 0 8])>> grid>> ylabel('Magnitude')>> subplot(212)>> plot(w,angle(H))>> axis([0 2*pi -pi pi])>> grid>> ylabel('Phase (rad)')Hejˆ ej2ˆ–ej2ˆejˆ3 ejˆ–ej2ˆ–+++ +=ej2ˆ–22ˆ2ˆ 3+cos+cos=Hejˆ 32 ˆcos 2 2ˆcos++=Hejˆ 2ˆ–=Sinusoidal Response of FIR SystemsECE 2610 Signals and Systems 6–5>> xlabel('hat(\omega) (rad)')Example: Find for Input • The input frequency is rad, the amplitude is 5, andthe phase is • Assuming is input to the 4-tap FIR filter in the previousexample, the filter output is• The amplitude response or gain at is =3.248; why?0 1 2 3 4 5 602468Magnitude0 1 2 3 4 5 6−202Phase (rad)hat(ω) (rad)–223.248-2ynxn 5ej 1 n=ˆ01= 0=xnyn 32 1cos 2 2 1cos++ej 12–5ej 1 n=16.2415ej2–ejn=ˆ01=Hej1Superposition and the Frequency ResponseECE 2610 Signals and Systems 6–6Superposition and the Frequency Response• We can use the linearity of the FIR filter to compute the out-put to a sum of sinusoids input signal• As a special case we first consider a single real sinusoid(6.7)• Using Euler’s formula we expand (6.7)(6.8)• The filter output due to each complex sinusoid is known from(6.3), so now using superposition we can write(6.9)• We can simplify this result to a nice compact form, if wemake the assumption that the FIR filter has real coefficients• Special Result: It will be shown in a later section of thischapter that an FIR filter with real coefficients has conjugatesymmetry(6.10)– What does this mean?xn A ˆ0n +cos=xnA2---ej ˆ0n +A2---ej– ˆ0n ++=ynA2---Hejˆ0ej ˆ0n +A2---Hej– ˆ0ej– ˆ0n ++=Hejˆ–H*ejˆ=Hejˆ–ReHejˆjImHejˆ–=HejˆHejˆ–=Superposition and the Frequency ResponseECE 2610 Signals and Systems 6–7• We now use (6.10) to simplify (6.9)(6.11)• We see that when a real sinusoid passes through an LTI sys-tem, such as an FIR filter (having real coefficients), the out-put is also a real sinusoid which has picked up the magnitudeand phase of the system at • The generalization (sum of sinusoids) of this result is when, (6.12)then the corresponding LTI system output is(6.13)ynA2---Hejˆ0ej ˆ0n