Contoh DSP - DFT Solved

Contoh 1

Verifikasi teorema Parseval dari barisan $ x (n) = \ frac {1 ^ n} {4} u (n) $

Solution- $ \ displaystyle \ sum \ limit _ {- \ infty} ^ \ infty | x_1 (n) | ^ 2 = \ frac {1} {2 \ pi} \ int _ {- \ pi} ^ {\ pi} | X_1 ( e ^ {j \ omega}) | ^ 2d \ omega $

Kiri $ \ displaystyle \ sum \ limit _ {- \ infty} ^ \ infty | x_1 (n) | ^ 2 $

$ = \ displaystyle \ sum \ limit _ {- \ infty} ^ {\ infty} x (n) x ^ * (n) $

$ = \ displaystyle \ sum \ limit _ {- \ infty} ^ \ infty (\ frac {1} {4}) ^ {2n} u (n) = \ frac {1} {1- \ frac {1} {16 }} = \ frac {16} {15} $

RHS $ X (e ^ {j \ omega}) = \ frac {1} {1- \ frac {1} {4} ej \ omega} = \ frac {1} {1-0.25 \ cos \ omega + j0. 25 \ sin \ omega} $

$ \ Longleftrightarrow X ^ * (e ^ {j \ omega}) = \ frac {1} {1-0.25 \ cos \ omega-j0.25 \ sin \ omega} $

Menghitung, $ X (e ^ {j \ omega}). X ^ * (e ^ {j \ omega}) $

$ = \ frac {1} {(1-0.25 \ cos \ omega) ^ 2 + (0.25 \ sin \ omega) ^ 2} = \ frac {1} {1.0625-0.5 \ cos \ omega} $

$ \ frac {1} {2 \ pi} \ int _ {- \ pi} ^ {\ pi} \ frac {1} {1,0625-0,5 \ cos \ omega} d \ omega $

$ \ frac {1} {2 \ pi} \ int _ {- \ pi} ^ {\ pi} \ frac {1} {1,0625-0,5 \ cos \ omega} d \ omega = 16/15 $

Kita dapat melihat bahwa, LHS = RHS. (Karenanya Terbukti)

Contoh 2

Hitung DFT titik-N $ x (n) = 3 \ delta (n) $

Solution - Kami tahu itu,

$ X (K) = \ displaystyle \ jumlah \ batas_ {n = 0} ^ {N-1} x (n) e ^ {\ frac {j2 \ Pi kn} {N}} $

$ = \ displaystyle \ sum \ limit_ {n = 0} ^ {N-1} 3 \ delta (n) e ^ {\ frac {j2 \ Pi kn} {N}} $

$ = 3 \ delta (0) \ times e ^ 0 = 1 $

Jadi, $ x (k) = 3,0 \ leq k \ leq N-1 $ … Ans.

Contoh 3

Hitung DFT titik-N $ x (n) = 7 (n-n_0) $

Solution - Kami tahu itu,

$ X (K) = \ displaystyle \ jumlah \ batas_ {n = 0} ^ {N-1} x (n) e ^ {\ frac {j2 \ Pi kn} {N}} $

Mensubstitusi nilai x (n),

$ \ displaystyle \ jumlah \ batas_ {n = 0} ^ {N-1} 7 \ ​​delta (n-n_0) e ^ {- \ frac {j2 \ Pi kn} {N}} $

$ = e ^ {- kj14 \ Pi kn_0 / T} $ … Ans