Well sorry for the bad news but the circuit would never work....
1)The IR LED is reverse biased.....swap the anode and cathode.
2)The 4.7k resistor wouldn't allow any(useful) current through the LED.
Considering a working voltage of 1.5v for the LED and say 0.5v across BC547's CE,we have 5-(1.5+0.5)=3v across the resistor which would provide for 3/4.7 = 0.64ma......a more usable resistor would be 56 or 68 ohms which would provide ~40-50ma.
3)There's no base resistor for the BC547 and the output pin of the TSOP is internally pulled up so 5v would be applied to the base.So add a 4.7k resistor to limit the base current and potential transistor damage.
4)Where's the carrier?TheTSOP includes a demodulator so what you have at it's output pin is the data minus the 38 khz carrier. Simply sending this data through the IR LED won't work because the Tatasky unit needs a
38Khz modulated signal from it's remote.So you'll have to modulate the data with a 38khz carrier.
On second thoughts,the carrier filtered out by the TSOP should still be present albeit at a very low level.The BC547 would amplify this along with the data and hence it may work. You need to test though.
5)The transistor employed would invert the signal hence we need an inverter before it so that the final output remains unaltered.You may use one part of a CD4049 hex inverter- buffer.
Whew that's about it.Look at the attachment for a modified ckt.Actually two....one using cd4049 inverter,the other using a transistor inverter.