Implement the ideal low pass filter frequency

Question

Problem 1: Read section 2.5 in Oppenheim and Schafer and answer the following questions.

Part (a) State the conditions required for a system whose input-output behavior is described by a linear constant coefficient difference equation to be causal, linear, and time-invariant.

Part (b) What conditions on the a[k] coefficients guarantee that the system has finite impulse response?

Problem 2: Consider the system that implements the ideal low pass filter frequency response depicted in Figure 2.17 (page 44).

(a) Is it causal? Explain your answer.

(b) Is it stable? Explain your answer.

Problem 3: Assume that x[n] ↔ X(f ) are a DTFT pair. Prove the DTFT symmetry properties:

(a) x[-n] ↔ X(-f )

(b) x*[n] ↔ X*(-f )

(c) x*[-n] ↔ X*(f )

Problem 4: Assume that x[n] ↔ X(f ) form a DTFT pair. Prove the following:

(a) x[n] ∈ R implies that X(f ) is Hermitian, X(f )= X*(-f )

(b) x[n] ∈ jR (pure imaginary) implies that X(f ) is anti-Hermitian, X(f )= -X*(-f )

(c) x[n] is even implies that X(f ) is even, X(f )= X(-f )

(d) x[n] is odd implies that X(f ) is odd, X(f )= -X(-f )

Problem 5: Let x[n]= 0.9ⁿej^Πn/4 u[n] and let y[n] = real(x[n]). Do the following on paper.

(a) Compute X(f ).

(b) Show that Y (f )= 1/2[X(f )+ X*(-f )], i.e. Y (f ) is the Hermitian part of X(f ).

Do the following in Matlab. Use enough points in your plot so that the curves appear smooth. Try 1000 points, for example. Turn in plots and your code.

(c) On one axis, plot |X(f )| and |Y (f)| over f ∈ [-1, 1] (i.e. two periods). What type of symmetry do you see in |Y (f)|?

(d) On one axis, plot ∠X(f ) and ∠Y (f ) over f ∈ [ 1, 1] (i.e. two periods). What type of symmetry do you see in ∠Y (f )?

(e) On one axis, plot real(X(f)) and real(Y (f )) over f ∈ [ 1, 1] (i.e. two periods). What type of symmetry do you see in real(Y (f ))?

(f) On one axis, plot imag(X(f)) and imag(Y (f )) over f ∈ [ 1, 1] (i.e. two periods). What type of symmetry do you see in imag(Y (f ))?

Problem 6: Prove the DTFT properties in Table 2.2 (page 58).

(a) Property 2: x[n - d] ↔ X(f )e^-j2Πfd

(b) Property 3: x[n]e^j2Πf0n ↔ X(f - f₀)

(c) Property 5: nx[n] ↔ j/2Π dX(f)/df

(d) Property 6: x[n] * y[n] ↔ X(f )Y (f )

(e) Property 7: x[n]y[n] ↔ X(f ) ⊗ Y (f )= ₀∫¹ Y(λ)X(f - λ)dλ

(f) Property 9: ∑^∞_n=-∞ |x|[n]y*[n] = 0∫1X(f)Y *(f )df

(g) Property 8 (use property 9): ∑^∞_n=-∞ |x|[n]|² = ₀∫¹|X(f)|2df

Problem 7: Prove the DTFT pairs in Table 2.3 (page 62). In each case state the type of signal x[n] (absolutely summable (stable), energy, or power) and comment on the attributes of the DTFT X(f ) (continuous, contains discontinuities, or contains impulses).

(a) Pair 1: δ[n] ↔ 1

(b) Pair 2: δ[n - d] ↔ e^-j2Πfd

(c) Pair 3: 1 ↔ ∑^∞_n=-∞ δ(f - k)

(d) Pair 4: aⁿu[n] ↔ (1 - ae^-j2Πfd)^-1

(e) Pair 5 (we will do this one in class): u[n] ↔ U (f )= (1 - e-^-j2Πfd)^-1 + 1/2 ∑^∞_n=-∞ δ(f - k)

(f) Pair 6 (use the derivative property): (n + 1)aⁿu[n] ↔ (1 - ae^-j2Πfd)^-2

(g) Pair 8: sin(Πn[2fc])/Πn ↔ u(f + f_c) - u(f - f_c)

(h) Pair 9: u[n] - u[n - M - 1] ↔ sin(Πf (M+1))/sin(Πf) e^-jΠfM

(i) Pair 10: ej^2Πf0n ↔ ∑^∞_k=-∞ δ(f - f₀ - k)

(j) Pair 11: cos(2Πf0n + Φ) ↔ 1/2 ∑^∞_k=-∞ ejΦδ(f - f₀ - k)+ e-jΦδ(f + f₀ - k)

Problem 8: For Pair 4 in which x[n]= anu[n], suppose a ∈ R and |a| < 1, i.e. x[n] is a real-valued sequence.

(a) Show that X(f ) has Hermitian symmetry by computing the real and imaginary parts of X(f ) and showing that the real part is an even function and the imaginary part is an odd function.

(b) Compute the magnitude and phase of X(f ) and show that the magnitude is an even function and the phase is an odd function.